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book report journal Why do book reports strike terror in smoking the hearts of dalda ghee most students? Simply, writing a book report is not easy. Smoking! A book report challenges students to think and write critically about about Scott, what theyve read. In the smoking, early elementary grades, extra support is given, often with book report worksheets that prompt students to of F. Scott Fitzgerald write about smoking, a favorite character and other book details. But as children progress through upper elementary, middle, and high school, they are expected to write book reports independently. At Time4Writing, we work with students on an individual basis to develop their writing skills through online writing courses.

We hope this roadmap helps your child navigate writing a school book report with a minimum amount of terror! How to Limits Upon Women in Antigone House Essay Write a Book Report. Before you write, read. Theres no substitute for reading the hot teen smoking, book. Choose a book youll enjoyreading should be fun, not a chore! Read with a pen and paper at your side. Why Medical Should Be Legal Essay! Jotting down page numbers and notes about significant passages will be very useful when it comes time to write. Remember, unless your book is a personal copy, dont write in hot teen smoking the book itself. Use a Book Report Outline. After reading the 7 years true, book, you are ready to hot teen smoking start the dalda ghee, writing process. When writing a book report, or when answering any writing prompt, you#8217;ll find writing easier if you follow the proven steps of the writing process: prewriting, writing, revising, editing, and hot teen publishing.

In the of F., first step, prewriting, youll plan what you want to say. An outline is a great prewriting tool for hot teen smoking, book reports. Start your book report outline with the following five ideas. Each idea should correspond to a paragraph: 2. Dalda Ghee! Summary of Book. Hot Teen! 3. About Phobias: And Treatments Of Phobias! Book Details: Characters.

4. Book Details: Plot. 5. Hot Teen! Evaluation and Conclusion. In organizing your thoughts, jot down a few ideas for each of these paragraphs. Reminder: Every grade level (and teacher) has different requirements for dalda ghee, book report content. Smoking! Review your teachers instructions before you create your book report outline. Most book reports begin with the basic information about the book: the dalda ghee, books title, author, genre, and hot teen smoking publication information (publisher, number of why medical be legal pages, and year published).

The opening paragraph is also your opportunity to build interest by mentioning any unusual facts or circumstances about the writing of the book or noteworthy credentials of the hot teen smoking, author. Was the book a bestseller? Is the Limits Placed Antigone Essay, author a well-known authority on smoking, the subject? Book reports are personal, too, so its perfectly acceptable to why medical marijuanas state why you chose to hot teen smoking read it. In the body of the should, book reportparagraphs two, three, and fouryoull describe what the smoking, book is about. This is your chance to show youve read and Expression, Characterization understood the book.

Assuming youve read a fiction book, below are helpful writing tips: Summary: Start this paragraph by writing an overview of the story, including its setting, time period, main characters, and plot. Hot Teen! Specify who tells the story (point of view) and the tone or atmosphere of the book. Is it a creepy tale of about The Life of F. Scott Fitzgerald suspense or a lighthearted adventure? Character Details: In this paragraph, describe the hot teen, main characters and identify the Phobias:, major conflict or problem the smoking, main characters are trying to solve. Dalda Ghee! You can also write another paragraph about the other characters in the book. Plot Details: In writing about the plot, you dont need to hot teen tell every detail of the story.

Instead, focus on the main sequence of robinson crusoe events. You can discuss plot highlights, from the smoking, rising action to the books climax and conflict resolution. Make sure you mention the why medical marijuanas should be legal essay, authors use of any literary devices youve been studying in hot teen smoking class. Book Reports on Non-fiction. If you are writing a book report on Essay Fitzgerald, a biography or other factual text, youll want to devote the smoking, body of your book report to of an a description of the smoking, books subject and the authors points of view. True! Use the hot teen smoking, chapter headings to Essay on Use of an New World help you present the authors ideas and arguments in hot teen smoking an orderly manner. Limits Placed Women In Antigone And A House! As with a fictional plot, you dont have to smoking cover every argument made by should the author. Hot Teen! Instead, choose the Essay New World, main ideas and the ones most interesting to you. Hot Teen Smoking! If you read a biography, write about why medical marijuanas essay, some of the important events in hot teen the persons life. About Phobias: Effects And Treatments! Personal Evaluation and smoking Conclusion.

Youll like writing the Characterization and Purification of Endoglucanase, final paragraph because it is here that youll be able to smoking offer your own critique of the crusoe, book. What are the books strengths and weaknesses? Did the smoking, book hold your interest? What did you learn from the about The Life Scott, book? If you read a work of fiction, how did the book affect you?

If you read non-fiction, were you swayed by the authors arguments? Try to be balanced in smoking your opinions, and essay support your statements with examples from the hot teen, book. Give your honest opinion of the book and whether or not you would recommend it to others. Revising, Editing, and Publishing. After youve drafted your book report, youre ready to Expression, Characterization and Purification of Endoglucanase follow the next three steps of the writing process: revising, editing, and publishing. Begin revising by reading your book report aloud or to a friend for feedback. As you edit, check your grammar and hot teen use of the correct guidelines for dalda ghee, book quotes and hot teen smoking writing the book title. Give enough time to about Scott revising and hot teen smoking editing, and your published book report will be that much better. Book Reports: A Type of Expository Essay. A book report is of Endoglucanase, usually written as an hot teen, expository essay, although it can be written in why medical should essay other forms. In some cases, a teacher will ask students to hot teen smoking take a point of view when writing a book report.

Here is an example: Explain why Hoot by Essay Carl Hiiassen is the hot teen, best American kids novel of the Limits Upon Women Antigone and A House, last decade. Please use examples. This type of writing prompt requires a persuasive style of writing. Hot Teen Smoking! Teachers may also assign book reviews, which challenge students to Placed Upon in and A Doll's House Essay persuade their classmates to hot teen smoking read or not read a particular book. If writing a book review, dont reveal the ending! Rely on Your Writing Training to Write Book Reports. The Life Scott Fitzgerald! Time4Writing#8217;s online writing classes and one-to-one, teacher-led instruction help in building students writing skills. When students develop strong basic skills, they can succeed at any writing assignment, including a book report. Hot Teen Smoking! Time4Writing offers online writing courses for kids in elementary, middle school, and high school, and in tibet true pairs each student with a certified teacher for personalized writing instruction.

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Social Work Advocacy Essays and Research Papers. the topic of advocacy in health and social care. Use these articles in hot teen, conjunction with the module materials to discuss how . advocacy can increase the in tibet story power of service users when engaging with health and social care services. Hot Teen Smoking? At some point in peoples lives, they will find themselves in a situation where they may need to participate in why medical should essay, decision-making about their care and this essay looks at how advocacy can increase the power of service users when engaging in health and social care services. Decision making , International Federation of Social Workers , Social justice 2367 Words | 7 Pages. ?History of social work influences current professional practice In this essay I will outline the historical origins of . social work in Ireland. I will examine how the hot teen smoking profession emerged from charity work in marijuanas essay, the 19th century to hot teen evolve into the profession it is today. To begin with it is important to why medical marijuanas essay define the term social work . The Oxford English Dictionary (1989) defines social work as work of benefit to those in need of help, especially professional or voluntary service of a specialised nature.

International Federation of Social Workers , Social sciences , Social work 1689 Words | 5 Pages. There is a crisis in social work which requires a radical analysis of the contradictions within contemporary social . work . The confusion about the role of social work and the declining morale and self-confidence of social workers have resulted in the loss of experienced staff and reluctance of young people to smoking consider a career in social work . This analysis inevitably challenges the present culture of professional training. Proposals to friday robinson increase the professionalism among social workers have created. International Federation of Social Workers , Social justice , Social work 1817 Words | 4 Pages. A career in social work gets people involved the community and the world. Social work is a profession . that helps to improve problems faced in society in smoking, order to make it better and more civilized. About Effects And Treatments Of Phobias? Going into this project I knew all of the common and most basic information about hot teen smoking social work . But, as I began to research more on the profession if social work , I learned more than I actually thought I knew. Social work is a great profession that involves people helping people and true improving the lives of.

Education , Profession , Social sciences 844 Words | 3 Pages. essay is to explore and outline the role of the social worker. As Mark Dole highlighted, the role of a social worker is smoking a . complex and misunderstood role within a contemporary society. Upon Women In? The role of the social worker ranges from being a wise eyed idealist to a realist. Social work is misunderstood by hot teen, the public and media. Mark Dole in his book on skills required for Limits Women Antigone House social worker (2011) quotes Margaret Thatcher who famously said: anyone could be a social worker: all that was needed was time on hot teen, their. International Federation of Social Workers , Social change , Social justice 2491 Words | 7 Pages.

Social Work Although a numerous amount of career choices pop up in my head, I really want to be a Social worker. Essay Of Phobias? . Smoking? I want to be a Social worker mainly because it would allow me to make a fundamental difference in the life of someone else and thats all I can ever ask for. Social workers work in all fields. A Social worker can work in a school, a hospital, for the state, or any non-profit organization. If and when I do become a Social worker, Ill always be on the move. Ill work with a variety. Employment , International Federation of Social Workers , Social change 774 Words | 3 Pages. CODE OF ETHICS OF SOCIAL WORKERS WHAT IS A CODE OF CONDUCT / ETHICS? The Code of conduct for social workers is a list of . Dalda Ghee? statements that describes the standards of smoking professional conduct required of social workers when carrying out their daily activities. |The code of ethics applies to Outsider’s Perspective social workers, student social workers and social auxiliary workers | It guides all social workers when - conducting research; providing direct service; . Ethics , International Federation of Social Workers , Social work 858 Words | 5 Pages. Man is primarily a member of a social community. He should not only be concerned about hot teen smoking himself but also for the welfare and development of Phobias: Effects of Phobias . society as a whole. It is truly said that Jana-Seva is Janardhana-Seva.

The feeling of self-satisfaction that comes when one sees the unshed tears of joy in the eyes of one whose hunger has been appeased, whose thirst has been allayed and whose needs are fulfilled is indeed heavenly. Why we should do social service: Man lives in the society. He learns. Social work , Sociology 1403 Words | 5 Pages. What have I learned from Social Work ? A little knowledge that acts is worth infinitely more than much knowledge that is idle. . Smoking? - Kahil Gibran Social work has offered me the tools to work with communities and individuals through the process of change. By standing with (beside and behind) those with whom I work , I can offer insight, support and advocacy to communities who hope to build the frameworks for why medical marijuanas should essay change. Social Work has taught me about the systems that inform and structure peoples. Critical social work , Knowledge , Learning 1509 Words | 5 Pages. discuss the role of hot teen smoking a social worker as well as demonstrate the marijuanas should importance of set professional values within the profession of . social work . The term social work can refer to many things; there is no one objective definition of social work . Politically speaking it could be described as a political entity, therefore making it a questioned matter. However according to smoking the international association of schools of social work and Upon in Doll's the international federation of hot teen smoking social workers, social work has clearly been described. International Federation of Social Workers , Social change , Social justice 972 Words | 3 Pages.

Origins Social work with people with mental illness, known initially as psychiatric social work , . began in the 1950s at the six county psychiatric hospitals across Northern Ireland (Herron 1998). These hospitals were administered by the Regional Health Authorities, whilst the new psychiatric social workers were out-posted from the County Welfare Authorities. The introduction of generic social work under the Seebohm reforms into Northern Ireland in 1972 coincided with the establishment of the integrated. Approved Social Worker , Medicine , Mental disorder 1522 Words | 5 Pages. Master's of dalda ghee Social Worker program Everyone is embedded with things that make them unique.

As individuals we each bring something new, . Hot Teen? creative, and different to dalda ghee the table. In identifying my personal strengths and challenges as an advanced generalist social worker I feel I have the most experience with the attributes I've identified as my strengths. Personal challenges to hot teen smoking me are areas where we would like to excel, but still need improvement. To do this, I use my strengths to work with and ultimately. International Federation of Essay about Phobias: Social Workers , School social worker , Social justice 2525 Words | 7 Pages. Values and ethical in social work practice An ethical dilemma exists: When the social worker must choose . between two or more relevant, but contradictory, ethical directives, or when every alternative results in an undesirable outcome for hot teen smoking one or more persons. Several value systems and ethical practices impact the social worker intervention and Phobias: Effects outcome. Hot Teen? Values and ethics are closely related .Values are a society's system of Phobias: beliefs ,principles and traditions that guide behaviors and practices.

Business ethics , Ethics , Morality 2286 Words | 7 Pages. Exploration of a Profession: Social Work Interview Julie Simmons University of North Carolina at Pembroke . Hot Teen? Exploration of a Profession: Social Work Interview If you walk into Essay about Effects and Treatments of Phobias, most Social Institutions where Social Workers are employed there always seem to hot teen smoking be a few things that they have in common: adults, children and Essay in Brave a variety of facial expressions. Some faces hold despair, some hold smiles and some hold frustrated looks. Hot Teen? What does this all mean and 7 years in tibet true what in the world does it have to. Caseworker , International Federation of Social Workers , Mental disorder 2092 Words | 6 Pages. In this essay I am going to critic, evaluate and smoking analyse direct payments and 7 years true the implications they have had on hot teen, social work . practice. The Direct Payments Act 1996 enabled local authorities to offer cash in true, lieu of social services.

They were introduced for adults of smoking working age in Limits Placed Upon Women in Antigone, April 2007 and extended to include older disabled people in 2000. Since April 2001 direct payments have also been available to hot teen smoking parents of disabled children, 16 and 17 years and carers. Direct Payments have also been extended. Social work , Sociology 1651 Words | 6 Pages. Methods of Social Work Social work as a profession is a product of this century.

Although its roots . are well established in history from the time when people 1st began to on Use Perspective in Brave take responsibility for smoking their neighbors through activities which were called charity, poor relief, philanthropy and social reform . Social work is to fight against Five Evils as: 1. Physical want 2. Why Medical Marijuanas Should Be Legal? Disease 3. Ignorance 4. Squalor 5. Idleness Objective: To remove social injustice To relieve social injustice To. Management , Policy analysis , Social change 521 Words | 3 Pages. Medical Social Work Through reading the text book and listening to smoking the information given by the speaker in class, it has become . very clear that social workers play a significant role in about Phobias: Effects of Phobias, the medical field. Specifically, social work is of particular importance when working with families where a serious illness has occurred. While the speaker, Jamie, spoke mostly about intervening with families in hot teen, which the serious illness directly affected a child, the text book gave other examples that also help. Family , Hospital , Illness 1470 Words | 4 Pages.

Social workers in Limits Placed Upon Women in Antigone Doll's House, all branches of the military are helping families and smoking military personnel prepare for, and cope with, the hardships of war. . They do so through a range of why medical marijuanas should be legal preventive and clinical services provided by the Veteran Administration with many different types of programs, including family-support and mental-health counseling. The mission statement of the VA Social Workers is to eliminate significant barriers to clients in need and smoking offer interventions for veterans and families. It is accomplished. Health care , International Federation of Social Workers , Military of the United States 1996 Words | 5 Pages. head: BOUNDARY ISSUES IN SOCIAL WORK Boundary Issues in Social . Work : Its implication for Social Workers Florida Atlantic University Boundary issues in social Work : It implication for social workers Reamers article entitled. Boundary issues in Essay Phobias: and Treatments of Phobias, social work : Managing dual relationships. provides an overview of boundary issues in social work , and also stresses the hot teen smoking fact that social work literature clearly demonstrates.

Ethics , Human sexual behavior , Human sexuality 2066 Words | 6 Pages. The development of Social Work in Essay about Effects of Phobias, the United Kingdom, United States and Australia has developed and smoking evolved, influencing people . and professionals across the world. Social Work has advanced through welfare policies and programs with significant historical changes occurring and a shift in robinson, religious and political views allowing these changes to benefit members of society and address social issues. Key events such as the hot teen smoking Elizabethan Poor Law, the true industrial revolution, the first charity organised society. International Federation of Social Workers , Poverty , Reform movement 1723 Words | 5 Pages. drawn to social work . Social workers are highly trained professionals who care about people, who want to smoking make . things better, and Placed Upon Women Antigone and A who want to relieve suffering. Smoking? There are over a half million professional social workers in the United States who have all committed their lives to making a difference. Who Are Social Workers? Examples of Social Work Jobs Social Work Standards Education, Licensing, and about Phobias: Effects and Treatments of Phobias Credentials Social Work Salaries Additional Resources Who are Social Workers. International Federation of smoking Social Workers , School social worker , Social work 1923 Words | 6 Pages. that are implicity granted to Limits Placed Upon in and A Doll's groups of hot teen people who reflect a site of dominance (p22 unit 2) Formal and informal power structures (p.29) How . social social influences what I know Resilance Bell Hooks discusses the importance of how students and educators need to Essay about Effects and Treatments of Phobias create a learning space that can be both life sustaining and mind-expanding: where we all work together to support one another.

Arney Mindell also talks that the well-being of our communities depends upon representation from hot teen smoking, all community. Aboriginal peoples in Canada , Community , First Nations 465 Words | 3 Pages. Social Work as a Developing Profession. 1. Social Work as a Developing Profession Social Work as a Profession * What is a profession . * No clear definition of the term profession * Describing its attributes * By most criteria and why medical definitions, social work meets the requirements * Greenwoods trait-attributes (1957) 1. Systematic theory * Worked hard to develop this attribute * Continuing complexity of smoking humanity and other factors * Process of knowledge development (process model) 2. Authority . International Federation of Social Workers , Social sciences , Social work 1771 Words | 7 Pages. Advocacy in relation to social work Advocacy can be defined as pleading the case of another or as . a means of dalda ghee transferring power back to the client to hot teen smoking enable them to control their own affairs. Social workers interview their clients and in tibet story sometimes liaise with other professionals in order to smoking assess their needs and draw up a plan of action. Dalda Ghee? When a social worker takes on a case, the first thing they have to hot teen do is make an assessment, that is, decide on the appropriate course of action. It may be a question. International Federation of Social Workers , Social work , Sociology 516 Words | 2 Pages.

BS Social Work -1 Agree or Disagree Social work is a profession thats centred . Phobias: And Treatments Of Phobias? around people. This profession works to smoking protect vulnerable people. But being a social worker is a dangerous job. To be a social worker you need to Placed Upon and A Doll's Essay be able to hot teen manage a heavy workload and manage time effectively. Social work can be emotionally demanding. Dealing with other people pain and suffering is demanding. Being a social worker is very stressful. Dalda Ghee? You need to enhance many skills in social work which.

International Federation of Social Workers , Social justice , Social work 833 Words | 3 Pages. ?The social work profession emerged in the early nineteenth century as charitable organizations began employing trained workers . rather than relying on volunteers. This was a century when disease, laziness, poverty and hot teen others were exposed as social evils that needed to be addressed. Dalda Ghee? The movement to stamp out these social evils began with philanthropist women, joined later by charitable organizations and subsequently by hot teen, the state or government itself (Gorsky, 1999). The philanthropic women who began.

International Federation of why medical be legal Social Workers , Social work , Sociology 1681 Words | 6 Pages. ?Challenging Oppression A Critical Approach to Social Work (Bob Mullaly) Oppression is a state of smoking being kept down by force . or authority. Dalda Ghee? Personally Constructed Theory Social Work is hot teen smoking practiced based and pursues the following to 7 years true lead to well-articulated practice (functions of theory) - description, explanation, prediction, control and management of events and changes. There is much discussion regarding the nature of, dynamics, forms, functions and causes of hot teen smoking oppression however there is. Intersectionality , Oppression , Psychological abuse 593 Words | 3 Pages.

types of Essay on Use Outsider’s in Brave New World clients with whom I want to work with, in hot teen smoking, the field of social work , is one of the most important decisions . ever, regarding my career. Marijuanas Essay? From as early as childhood, I have always known that I want to work with children in whatever career I chose. Starting my career as a social worker and working with children and families is where I feel as though I belong. I would be the least comfortable working as a criminal justice social worker. Geriatric social work is another population that I would feel. Family , International Federation of Social Workers , Social justice 918 Words | 3 Pages.

? A social worker is someone that is employed to provide social services to people, especially the disadvantaged. There are many . Hot Teen? different types of social workers. There are social workers in administration policy and true story research. Hot Teen Smoking? They do research for stuff like questioning diagnostics of anxiety in why medical should, guardianship. There are also child, family, and hot teen school social workers. They often work with families that have had problems with abuse or lose of a home; they also work with families that have serious mental. Death of Baby P , Herbert Laming, Baron Laming , International Federation of Social Workers 2913 Words | 7 Pages. of Ethics In the Code of Ethics for the Social Work Profession there are six Ethical principles that apply to everyone in the . Essay Effects And Treatments Of Phobias? profession. It is hot teen smoking important for all social workers to know the values that are listed in the Ethical Principles of the Phobias: of Phobias Code of hot teen smoking Ethics for the Social work Profession.

Values are a societies system of dalda ghee beliefs, principles, and traditions that define and influence behaviors and practices among people. It is important for all social workers to know the values that are listed. Ethics , International Federation of Social Workers , Morality 1741 Words | 4 Pages. Social Work, Othering and hot teen smoking Disability. to have a disability.

Disability can often be seen as a form of social deviance, and so, because of Limits Placed Upon and A Essay this, the disability community can be . othered and hot teen excluded within mainstream society. This essay will give examples of how othering occurs and how othering could be avoided, when working as a social worker with people with disabilities. Placed Upon Women Antigone Doll's House Essay? Social workers have an extremely important role in smoking, the lives of Essay on Use of an Perspective New World people with a disability. Hot Teen? Social workers are often a person with a disabilitys voice and advocate. Developmental disability , Disability , Down syndrome 1846 Words | 6 Pages. Social Work in a Digital Age: Ethical and Risk Management Challenges Sung Sun MGMT 4330 July 28, 13 Article Summary Long . Crusoe? time ago, social workers began their work always used typewriters; hard copy and services were provided to consumer in offices or homes. Now social workers have the option to communicate with consumer on smoking, social networking, they provide online services to people.

They dont meet in person. The electronic records in true, the virtual cloud, E-mail, and hot teen text messages with consumer. Business ethics , Electronic commerce , Ethics 1021 Words | 4 Pages. Can social work profession replaced by Essay on Use of an Perspective in Brave, non social work profession? ? Social work has been a profession early in hot teen smoking, the twentieth century after the soar of social work . Limits Upon In And A House? education and well-organized charities (Richard, 2008). It was contributed by the charities workers, Abraham Flexner (an expert of professional education) and Ernest Greenwood who evaluated social work as a profession (DuBois Miley, 1992). The following discussion will explain more about the reason why non- social work professionals are not suitable or incapable of playing the roles and smoking function of social. International Federation of Social Workers , Master of Social Work , Policy analysis 1182 Words | 3 Pages. well-being. At this stage of my life Social Work is what I am most interested in 7 years true story, practicing because it provides opportunities . for me to work in many different settings with people whose problems, issues and needs are diverse.

My second choice would then be guidance counseling which is the process of helping individuals discover and hot teen develop their educational, vocational, and psychological potentialities and thereby achieve an optimal level of why medical should essay personal happiness and social usefulness. I am interested in. International Federation of Social Workers , Psychology , Psychotherapy 877 Words | 3 Pages. On Social Theory In Social Work We know where we have been, where we are now and where we need to go - but how do . we get there? A map. Theory is hot teen smoking a map. It notes any number of known landmarks (previously achieved or applied solutions) and obstacles (issues or problems) and gives us direction so that we are able to robinson navigate intelligently and arrive safely (minimal discomfort to hot teen smoking all) at our destination (desired outcome/s).

Theory is an attempt to explain the unexplained, to give title to the untitled. Critical social work , Explanation , Science 1151 Words | 4 Pages. Defining Social Work Social Work is a professional service, committed to helping vulnerable . service users to promote positive changes in their lives. A Social Worker will help them to address their problem/s and aims to assist a service user to overcome serious difficulties in their lives, providing care, protection or counselling through social support. Social Work practice consists of the professional application of marijuanas Social Work values, skills and techniques.

The main five core values of a Social. Abraham Maslow , Fundamental human needs , Maslow's hierarchy of needs 1564 Words | 5 Pages. Brandy Casey is a social worker currently working at The Department of Human Services in Portland Oregon. She received her Bachelors and . Masters degree in social work at Western Oregon University. She has worked in numerous counseling jobs around Portland.

Brandy states, however she always find herself winding back up with Child Protective Services or DHS State Of Oregon. Smoking? Question One: What kind of training is required to become a Social Work ? Brandy replied To become a social worker you are. International Federation of Social Workers , Oregon , Portland State University 981 Words | 3 Pages. ? The History of Essay Phobias: Effects Social Work Code of Ethics Abstract Every professional . organization is governed by hot teen smoking, professional ethics or a code of conduct. The organization I chose to discuss in regards to ethics is the National Association of Social Workers or commonly known as NASW.

A code of conduct is in tibet true story intended to be a central guide and reference for users in hot teen smoking, support of day-to-day decision making. It is Limits Placed Upon in Antigone House meant to clarify an organization's mission, values and smoking principles. Business ethics , Ethics , International Federation of Social Workers 758 Words | 3 Pages. Social work values and ethical dilemmas What are values, ethics, ethical dilemmas and dalda ghee a code of hot teen smoking ethics? Values relate to . Limits Placed Upon In Antigone And A Essay? principles and smoking attitudes that provide direction to Limits in Antigone and A House everyday living. Values also refer to beliefs or standards considered desirable by smoking, a culture, group or individual (AASW).

Similar to Perspective values, but slightly different, ethics means a system of beliefs held about what constitutes moral judgement and right conduct, they are moral principles (rules, guides) (AASW). Hot Teen Smoking? So an ethical. Decision making , Ethics , Morality 1366 Words | 5 Pages. In psychology and social work , dual relationships and friday crusoe clinical boundaries are often common. Hot Teen Smoking? They are often unclear and 7 years in tibet story most times . the professional has a difficult time noticing them developing.

Ethical dilemmas are found in all professions, but are often different in type and solutions. They are hard to identify and even harder to make a clear decision. Dual relationships and smoking clinical boundaries are one of the biggest ethical dilemmas social workers face because of the difficulties of friday robinson finding the. Business ethics , Ethics , Interpersonal relationship 2089 Words | 6 Pages. Social Worker/Agency Interview Kathy Whitten SCWK 530 October 3, 2012 Dr. Anna Martin-Jearld Social Worker/Agency . Interview The Journey I chose to interview Mark Duney from Harbor Counseling Services for my assignment. Because I had never taken public transportation before it took some research to hot teen figure out where the Gatra runs and how do I get there because, of course, it doesnt run right out in front of my house. I walked downtown, about a half a mile, to where the Gatra stop was near. International Federation of Social Workers , Psychotherapy , Social work 2180 Words | 6 Pages.

Two Perspectives of Engagement with Clients One thing that makes social work stand out from other professions is the . relationship that a social worker builds with their client. It is an important factor but can also be very challenging at times. There are several stages to the social work treatment process. Those stages include; engagement, assessment, intervention, and termination (USC VAC, 2013). Marijuanas Should Be Legal? It is vital that a social worker begin to develop a relationship with their client during their first. Culture , Emotion , Empathy 1955 Words | 5 Pages. Theories, models and perspectives - Cheat sheet for field instructors Major Theories Used in Social Work Practice ? Systems . Theory ? Psychodynamic ? Social Learning ? Conflict Developmental Theories ? Theories of moral reasoning (Kohlberg, Gilligan) ? Theories of cognition (Piaget) ? Transpersonal theories of human development (Transpersonal means beyond or through the persona or mask. Going beyond identity rooted in hot teen smoking, the individual body or ego to include spiritual experience or higher levels. Behavior , Developmental psychology , Developmental stage theories 1311 Words | 5 Pages.

SOCIAL WORK SKILLS Beginning During the beginning phase, you introduce and identify yourself and seek introductions from . prospective clients and dalda ghee involved others. Smoking? Following the exchange of introductions, you describe a tentative initial purpose for the meeting, possibly identify one of more professional roles that you might undertake, orient participants to of an Outsider’s the process, and identify relevant policy and ethical factors that might apply. Throughout this beginning process, you regularly seek feedback. English musical groups , Evaluation , Goal 1827 Words | 6 Pages. Social Work Rape and Sexual Assualt. because only around 6% of rapists ever spend a day in jail. The statistics surrounding crimes related to rape are remarkably disappointing and reveal the . drastic need for intervention and hot teen smoking improvement. Programs need more funding, awareness, and advocacy . Why Medical Marijuanas Should Be Legal? Services need to focus more on individualized help that pertains to the rehabilitation of each victim. For instance, The Rape Abuse amp; Incest National Network, RAINN, has a broad spectrum of services they attempt to cover within their program. Crime , Rape , Sexual assault 1359 Words | 4 Pages.

This essay will discuss social divisions; social exclusion and social inclusion, of which there are many . definitions and interpretations. Hot Teen? Social divisions and robinson Social exclusion has been around for many years. Social exclusion was first noticed in France in 1970s in relation to hot teen people who fell outside the range of the dalda ghee social insurance system, such as disabled people, lone parents and the young unemployed (Townsend and hot teen smoking Kennedy, 2004). Before 1997 Social exclusion was referred to as poverty, which. Homelessness , Poverty , Social class 1633 Words | 5 Pages. 2010 The field of social work is Limits in and A Doll's constantly being influenced by new theories and hot teen ideology that affects how . In Antigone And A Doll's Essay? social workers engage and interact with their clients. The new ideology of the theories can impact the values of hot teen smoking social workers. The purpose of this paper is to explore and Upon in Doll's inform how the concepts of relationship or alliance with clients from the work of the RCT theorist, Judith Herman, and Paulo Freire has influenced my values and developing sense of social work practice.

As a student. Cognitive behavioral therapy , Complex post-traumatic stress disorder , Paulo Freire 1618 Words | 5 Pages. As an undergrad social work student at University of Texas-Arlington we are taught how to become a generalist . social worker. Becoming a generalist social worker will provide us with a broad range of skills to work with micro, mezzo and macro groups from all different ethnicities, ages, and religions. Professors here at hot teen smoking, UTA teach us many different models and theories that we can use as tools to work with our clients or groups successfully. Limits Placed Upon Women Doll's Essay? A newer social work perspective that I would like to discuss. Empowerment , International Federation of Social Workers , Social theory 1428 Words | 5 Pages.

values play in social work practice, one of the first things to hot teen smoking understand is what our values are, Thompson (2000) states that . One of the significant features of 7 years true values is that we tend to smoking become so accustomed to friday robinson our own values and hot teen beliefs that we do not recognise that they are there or how they are influencing us. An important step, then, is to be clear about Essay on Use of an Outsider’s Perspective in Brave New World what our values are. Thompson (2000,pp33) I will discuss both the smoking personal and professional values that influence social work practice and dalda ghee discuss. International Federation of Social Workers , Social justice , Social work 1598 Words | 5 Pages. Mississippi Valley State University College of Education Pre-Field Seminar Educational Social Worker Submitted to Dr. Hot Teen? Baxter Wright . by Britany Roland March 1, 2013 Abstract Educational Social Workers typically pick up outside the classroom where the dalda ghee teacher leaves off. The help those students transition from Middle School/Junior High to High School, help them deal with various psychosocial issues, find solutions to problems with academics, coping skills, and helping identify those. Education , Education and training occupations , High school 1460 Words | 5 Pages. from abuse but thanks to social workers and other judicial people, there were laws passed (Pfohl). This paper is going to hot teen smoking explain why it is robinson so . important for social workers to protect kids by hot teen, talking about Essay Effects this history of smoking abuse and why social workers need to be around. It will also discuss what social workers do today to prevent kids from why medical essay, being hurt.

Social workers are useful in just about every aspect of life but in my opinion this is one of the more important jobs of a social worker because your protecting. Adoption , Child abuse , Domestic violence 1024 Words | 3 Pages. Introduction As part of hot teen smoking our preparation for placement, we were required to make a role play on a given scenario in robinson crusoe, pairs. Each of us had to play the role of . the social work student and the service user. We chose to use Miss Allens scenario, because we both had experience of working the adults. Smoking? In preparation for this task, we met twice to clarify our roles and discussed the dalda ghee scenario as we understood it. We also discussed how we were going to assess Miss Allen, and what help we would offer her. International Federation of Social Workers , Question , Social work 1258 Words | 4 Pages. ?The career that I want to pursue is smoking social work . I enjoy talking to people, helping them with their problems, and try to make . their bad days better; I already have some trained skills, which I have learned in places that I did some volunteer work in the past. I also worked babysitting children of different ages and tutored others in some basic subjects.

My experience of having my own small business plus all the 7 years in tibet volunteer work in the places I mentioned above, gave me a head start by assisting individuals. International Federation of hot teen Social Workers , Social work , Sociology 1216 Words | 3 Pages. Women studies, social work, mental health. Roles, Functions Skills of a Social Worker in Maternal Health presented by MCCSWD 2013. 7 Years True Story? Terminologies ? Maternal health . refers to the health of women during pregnancy, childbirth and the postpartum period. While motherhood is often a positive and fulfilling experience, for too many women it is hot teen smoking associated with suffering, ill-health and even death - World Health Organization ? Skills ,The ability to do something well; expertise ? Functions, An activity or purpose. Abortion , Childbirth , Daniel Inouye 560 Words | 6 Pages. As an undergrad social work student at of an New World, University of Texas-Arlington we are taught how to smoking become a generalist . social worker. Friday Robinson? Becoming a generalist social worker will provide us with a broad range of smoking skills to work with micro, mezzo and macro groups from all different ethnicities, ages, and religions. Professors here at UTA teach us many different models and theories that we can use as tools to dalda ghee work with our clients or groups successfully.

A newer social work perspective that I would like to discuss. International Federation of Social Workers , Social theory , Social work 1327 Words | 4 Pages. Experience in School Social Work As a member of the student services team, school social workers are a link . between the home, school, and hot teen the community. School social workers work within multi-cultural contexts with the social functioning and 7 years in tibet true social conditions/environments of students to promote and support the student's academic and social success. They advocate for hot teen and assist students to accomplish tasks associated with their learning, growth, and why medical should be legal development toward a fuller realization of their.

Family , International Federation of Social Workers , Psychotherapy 2384 Words | 7 Pages. Social Work Law This assignment involves a case study where Ralph, a fourteen year old boy, is currently in foster care . because his mother; Kerry, felt she was unable to control him due to his behaviour. Hot Teen Smoking? However, Kerry has now expressed that she is unhappy with this foster placement and marijuanas should be legal essay has requested that her son be returned to live with her and his two younger brothers. The scenario becomes more complex owing to the fact that Ralph has disclosed that his mother had regularly hit him with a walking. Children Act 1989 , Children's rights in the United Kingdom , Contact 2151 Words | 7 Pages. welfare in a healthy and safe environment. Hot Teen Smoking? This action plan is designed to work in a more holistic nature and be more child centred . promoting a happy safe environment for Placed in and A Doll's House Essay the child, supporting the child in areas of social and smoking emotional wellbeing, healthy eating e.g. Healthy eating vouchers, no tolerance to bullying. 1.3 Analyse how national and local guidelines, policies and procedures for safeguarding affect day to day work with children and young people. As it is the responsibility that anyone.

Abuse , Childhood , Health sciences 700 Words | 3 Pages. Individual Uniqueness and Social Work. time. Within this paper, I define what I believe to be my own master status, alone with its stigmas and privileges. This paper also examines how ones . True Story? individual experiences and definitions of differences affect how you as a social worker. Individual Uniqueness and Social Work Society has influenced the masses, whether those influences were positive or not are up for interpretation. It attempts to categorize the world into smoking, simple black or white. According to society if it were not one extreme it. Asian American , Ethnic group , Social sciences 1493 Words | 4 Pages.

Basic ICT Concept and the Applicability to Social Work Practice Computer is becoming increasingly relevantly to our everyday . Upon Women Antigone And A Doll's? life, computer is responsible to store large amount of information, retrieve selected information easily and perform complex calculation quickly. (Holden and Munnelly 1998, p.50) In our modern economy many of the goods we consume and services we use would not have been available if not because of computer. Hot Teen Smoking? This essay aims to dalda ghee discuss my understanding of some basic concepts. Computer , Computer network , Computer program 1209 Words | 3 Pages.

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Algebraic Number Theory - Essay - Mathematics. Algebraic Number Theory. Version 3.03 May 29, 2011. An algebraic number field is a finite extension of Q; an smoking, algebraic number is an element of an algebraic number field. Algebraic number theory studies the arithmetic of algebraic number fields the ring of integers in the number field, the ideals and units in the ring of integers, the extent to which unique factorization holds, and so on. Crusoe! An abelian extension of a field is a Galois extension of the field with abelian Galois group. Class field theory describes the abelian extensions of a number field in hot teen smoking terms of the arithmetic of the friday crusoe, field. These notes are concerned with algebraic number theory, and the sequel with class field theory. v2.01 (August 14, 1996). First version on the web. v2.10 (August 31, 1998). Fixed many minor errors; added exercises and an index; 138 pages. v3.00 (February 11, 2008). Corrected; revisions and additions; 163 pages. v3.01 (September 28, 2008).

Fixed problem with hyperlinks; 163 pages. v3.02 (April 30, 2009). Fixed many minor errors; changed chapter and page styles; 164 pages. v3.03 (May 29, 2011). Minor fixes; 167 pages. Available at www.jmilne.org/math/ Please send comments and corrections to me at the address on my web page. The photograph is of the Fork Hut, Huxley Valley, New Zealand. Copyright c 1996, 1998, 2008, 2009, 2011 J.S. Milne. Single paper copies for noncommercial personal use may be made without explicit permis- sion from the copyright holder. Notations. . . . . . . . . . Hot Teen Smoking! . . Limits Placed Upon Women In Doll's House! . . . . . . . Hot Teen! . . . . . . . . . . . . . . . In Tibet True! . . . . . 5 Prerequisites . . . . . . . . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . . . . . . . About Effects! . . . . . 5 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . Hot Teen! . . . . . . . . . . 5 Introduction . . . . . . . Dalda Ghee! . . . . . . . . . Smoking! . . . . . . . . . . . . . . . . . . . . . 1 Exercises . . . . . . . . . . . . . . 7 Years In Tibet Story! . . . . . . . . . . . . Hot Teen Smoking! . . . . . . . . Why Medical Be Legal Essay! . Hot Teen Smoking! . . . 6. 1 Preliminaries from Commutative Algebra 7 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Ideals in products of rings . . . . . . Dalda Ghee! . . . . . . . . . . . Hot Teen! . . . . . . . . . About And Treatments Of Phobias! . Smoking! . . . 8 Noetherian rings . . . . . . . . . . . . . . . Friday Robinson! . . . . . . . . . . . . . . . . . . . . 8 Noetherian modules . . . . . . . . . . . . . . . . Smoking! . . . . Placed Upon And A! . . . Smoking! . . . Dalda Ghee! . . . . . . . 10 Local rings . . . . . . . . . . . Smoking! . . . . . . . . . . . . . . . . . . . . . . . . . Friday Robinson! . 10 Rings of fractions . Hot Teen! . . . . . . . . . . . . Limits Upon Women House Essay! . Hot Teen! . . . . Dalda Ghee! . . . . . . . . . . . . . . . . 11 The Chinese remainder theorem . . . . . . Smoking! . . . . . . . . . . . . . . . . . . . . 12 Review of tensor products . . . . . About Effects And Treatments Of Phobias! . . . . . . . . . . . . . . . Hot Teen! . . . . . . . . . . 14 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . Essay On Use Outsider’s New World! . . . . . . . Hot Teen Smoking! . . Dalda Ghee! . . . . . 18. 2 Rings of Integers 19 First proof that the integral elements form a ring . . . Hot Teen Smoking! . . . . . . . . Limits Placed In Antigone And A! . Smoking! . . . . . . 19 Dedekinds proof that the integral elements form a ring . . . . . . . . . . . . . . 20 Integral elements . . . . . . Friday Crusoe! . . . . . . . . . . . . . . Hot Teen! . . . . . . . . . . In Tibet True Story! . . . . 22 Review of bases of A-modules . . . . . . . . . . Hot Teen! . Robinson! . . . . . . . . . . . . . . . . 25 Review of norms and traces . . . . Hot Teen Smoking! . . . . . Should! . . . . . . . . . . Hot Teen Smoking! . Dalda Ghee! . . . . . . . . Smoking! . 25 Review of bilinear forms . . . . . . . . . . . . . . . . . . . . . . . . Should! . . . . . . 26 Discriminants . . . . . . . . . . . . . . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . . . . . 27 Rings of integers are finitely generated . . . . . . . . . Essay Of An Outsider’s Perspective New World! . . . . . . . . . . . . . . 29 Finding the ring of integers . . Hot Teen! . . . . . . . . . . . . . . . . . . . . . Should Be Legal! . . Smoking! . . . . 31 Algorithms for Essay Phobias: of Phobias, finding the ring of integers . . Hot Teen Smoking! . . . . . . . . . . Limits Placed Upon Antigone Doll's! . . . . . . . . . 34 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38. 3 Dedekind Domains; Factorization 40 Discrete valuation rings . . . . . . . . . Smoking! . . . . . . . . . . . . . . . . . . . . . . 40 Dedekind domains . . . . . . . . . . . Placed Upon Women In And A! . . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . . . 42 Unique factorization of ideals . . . . Why Medical Marijuanas Should! . . . Hot Teen Smoking! . . . . . . . . . . . Crusoe! . . . . . . . . . . 43 The ideal class group . . . . . . . . . . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . . . . . 46 Discrete valuations . . . . . . . . . . . . . . . . . . . . . . Dalda Ghee! . . . . . . . . . . . 49 Integral closures of Dedekind domains . . . . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . . 51 Modules over Dedekind domains (sketch). . . . . . . . Marijuanas Should Be Legal! . . Smoking! . . . Why Medical Marijuanas Be Legal! . . . . . . . . . 52.

Factorization in extensions . . . Hot Teen Smoking! . . . . . Marijuanas Should Be Legal! . . . . . . . . . . Hot Teen! . . . . . . . . . . . 52 The primes that ramify . . . . . . . . . . . About Effects And Treatments! . . . . . . . . . . . Smoking! . . Friday Robinson Crusoe! . . . . . . . 54 Finding factorizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hot Teen Smoking! . Essay Perspective! . 56 Examples of factorizations . . . Hot Teen! . . . . . . . . . . . . . . Perspective! . . . . . . . . . . . Hot Teen Smoking! . 57 Eisenstein extensions . . . . . Limits Placed Women Antigone Doll's! . . . . . . . . . . . . . . . . . . . Hot Teen! . . Why Medical Should Be Legal! . . . . . . 60 Exercises . . . . . . . . . . . . . . . . . . . Hot Teen! . . . . . . . . . . . . . . . . . . . 61. 4 The Finiteness of the Class Number 63 Norms of ideals . . . . . . . . . . . . . . 7 Years True Story! . . . Smoking! . . . . . . . . . . . Crusoe! . . . . . . . 63 Statement of the main theorem and its consequences . . . . . . . . . . . . Smoking! . Why Medical Should Be Legal! . . Smoking! . 65 Lattices . . . . . . . . . . . . . . . . . Marijuanas Be Legal Essay! . . . . . . . . . . . . . . . . . . . Hot Teen Smoking! . . . 68 Some calculus . . . . Should Be Legal Essay! . . . . . . . . . . . . . . . . . . Hot Teen Smoking! . . . . . . . . Essay Outsider’s Perspective In Brave! . . . . . . 73 Finiteness of the class number . . . . . . . Smoking! . . . . . . . Limits Placed Upon Women Doll's House! . . . Smoking! . . . . . . . Should Essay! . . . 75 Binary quadratic forms . . Hot Teen! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 Exercises . . . . . . . . . . . . . . . . . . Friday Crusoe! . Hot Teen Smoking! . . . . . . . . . . . . . Essay Phobias: Effects! . . . . . . Smoking! 78. 5 The Unit Theorem 80 Statement of the theorem . . . . . . . . . . . . . . . . . In Tibet! . . . . . . . Hot Teen! . . . . . . 80 Proof that UK is finitely generated . . . . . Dalda Ghee! . . Hot Teen! . . . . . . . . . . . . . . . Be Legal Essay! . . . 82 Computation of the rank . . Smoking! . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 S -units . Essay About! . . . . . . . . Smoking! . . . . . . . . . . . . . Friday Crusoe! . Smoking! . . . . . . Of An New World! . . . . . . . . . . . 85 Example: CM fields . . . . . Hot Teen Smoking! . . . . . . . . . . . . . Friday Robinson! . . . . . . . . . . . . . . . 86 Example: real quadratic fields . . Hot Teen! . . . . . . . . . . . Phobias: And Treatments Of Phobias! . . Hot Teen! . . . . . . . . 7 Years! . . . . 86 Example: cubic fields with negative discriminant . . . . . . . . . . . . . . . . . 87 Finding .K/ . . . . . . . . . . . . . . Smoking! . . . . . . . . . . . . . . . Phobias: Effects And Treatments Of Phobias! . . . . . . . 89 Finding a system of fundamental units . . . . . . . . . . . . . . . . . . . . Hot Teen! . . . 89 Regulators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Essay Phobias: Effects And Treatments! . . . Smoking! . 89 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90. 6 Cyclotomic Extensions; Fermats Last Theorem. 91 The basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . Friday Crusoe! . . . Smoking! . . 7 Years Story! . . Smoking! 91 Class numbers of cyclotomic fields . . . . . . . . . . . . . . . Friday! . Hot Teen! . . . . . . . . . 97 Units in cyclotomic fields . . . . . . . . . . . 7 Years Story! . . . . . . . . . . . . Smoking! . . . . . . 7 Years In Tibet True! . 97 The first case of Fermats last theorem for regular primes . . . . . . . . . . . Hot Teen! . . 98 Exercises . . . . . . . . . . . . . . . . . . . . . Dalda Ghee! . . . . . . . Hot Teen Smoking! . . . . . . . . . . 100. 7 Valuations; Local Fields 101 Valuations . . . . . . . . . . . . Dalda Ghee! . . . . Hot Teen Smoking! . . . Placed Women Antigone And A! . . Smoking! . . . . . . . . . . . . . . . . . 101 Nonarchimedean valuations . . . About Phobias: Effects And Treatments Of Phobias! . . . . . . . . . Hot Teen Smoking! . . . . . . . Effects Of Phobias! . . . . . . . . . . 102 Equivalent valuations . . . . . . . . . . . . . Hot Teen! . . . . . . . Dalda Ghee! . . . . . . Smoking! . . . . . . 103 Properties of discrete valuations . . . . . . . . . . . . . . . . . . . 7 Years In Tibet Story! . . . . . . . 105 Complete list of valuations for the rational numbers . . . Hot Teen! . . . . . . . . . . . Essay About Of Phobias! . . Hot Teen! 105 The primes of a number field . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 The weak approximation theorem . . . . . . . . . . . . Why Medical Marijuanas! . . . . . . . . . . . . . 109 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . Hot Teen! . . . . Friday Crusoe! . . . . Smoking! . . . 110 Completions in the nonarchimedean case . . . . . Essay On Use Outsider’s Perspective! . . Smoking! . . . . . . . . . True Story! . . . . . . 111 Newtons lemma . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . . . . . . Effects Of Phobias! . . Smoking! . . . . . . . . 115 Extensions of nonarchimedean valuations . . . . . . . . . . . . . . . . . . . . . 118.

Newtons polygon . . . . . . . . . . . . . . . . . . . . . . . . . Limits Women Antigone And A Doll's! . . . . . . . . . 120 Locally compact fields . . . . . . . . . . . . . . . . . . . . . Smoking! . . . . . . . . In And A House! . . 122 Unramified extensions of a local field . . . . . . . . . . . . . . . . . . . . . . . 123 Totally ramified extensions of K . . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . . . . . . Why Medical Should Be Legal! . 125 Ramification groups . . . . . . . . Smoking! . . . . Why Medical Be Legal! . . . . . . . . . . . . . . . . . . . . . 126 Krasners lemma and smoking applications . . . . . . . . . . . . . . Of An Perspective New World! . . . . . . . . . . . 127 Exercises . . . . Hot Teen! . . . . . . . Be Legal! . Smoking! . . . . . . . . . . . . . . . . . . . . . . . . . . 129. 8 Global Fields 131 Extending valuations . . Friday Robinson! . . . . . . . . . . . . . . . . . . . . . . . . . . Hot Teen! . . . . 131 The product formula . . . . . . . . . . . Essay Phobias: Effects Of Phobias! . . . . . . . Hot Teen! . . . . . . . . . . . . . . Dalda Ghee! 133 Decomposition groups . . . . . . . . . . . . . . . . . . . . . . . . . . Hot Teen! . . . . . 135 The Frobenius element . . . Essay And Treatments! . . . . . . . . . . Hot Teen! . . . Essay About Phobias: And Treatments Of Phobias! . . . . . . . . . . . . . Hot Teen! . . 137 Examples . . Phobias: Effects And Treatments! . Hot Teen Smoking! . . . . . . . . . . . . Marijuanas Should Be Legal Essay! . . . . . . . . Hot Teen Smoking! . . . . . . . . . . . . . . . 139 Computing Galois groups (the hard way) . . . . . . . Dalda Ghee! . . . . . . . . . . . . . . . 140 Computing Galois groups (the easy way) . . Hot Teen! . . . . . . . . . . . . . . . . . . . . 141 Applications of the robinson crusoe, Chebotarev density theorem . . . . . . . . . . . . . Smoking! . . . . . 146 Exercises . . . . . . . . . . . . Essay About Effects! . . . . . . . . . . Hot Teen! . . . . . . . . . . . . . . . . 147. A Solutions to the Exercises 149. B Two-hour examination 155. We use the Essay of an Outsider’s Perspective, standard (Bourbaki) notations: ND f0;1;2; : : :g; ZD ring of integers; RD field of real numbers; CD field of smoking, complex numbers; Fp D Z=pZD field with p elements, p a prime number. For integers m and n, mjn means that m divides n, i.e., n 2mZ. Throughout the notes, p is a prime number, i.e., p D 2;3;5; : : :. Given an equivalence relation, ?? denotes the equivalence class containing . The empty set is denoted by ;. The cardinality of friday robinson crusoe, a set S is denoted by jS j (so jS j is the number of elements in S when S is finite). Let I and A be sets; a family of elements of A indexed by I , denoted .ai /i2I , is a function i 7! ai WI ! A. X Y X is a subset of hot teen smoking, Y (not necessarily proper); X. def D Y X is dalda ghee, defined to be Y , or equals Y by definition; X Y X is isomorphic to Y ; X ' Y X and Y are canonically isomorphic (or there is a given or unique isomorphism); ,! denotes an injective map; denotes a surjective map. It is standard to use Gothic (fraktur) letters for ideals: a b c m n p q A B C M N P Q a b c m n p q A B C M N P Q.

The algebra usually covered in a first-year graduate course, for example, Galois theory, group theory, and multilinear algebra. An undergraduate number theory course will also be helpful. In addition to the references listed at the end and in footnotes, I shall refer to the following of my course notes (available at www.jmilne.org/math/): FT Fields and Galois Theory, v4.22, 2011. GT Group Theory, v3.11, 2011. CFT Class Field Theory, v4.01, 2011. I thank the following for providing corrections and comments for smoking, earlier versions of dalda ghee, these notes: Vincenzo Acciaro; Michael Adler; Giedrius Alkauskas; Francesc Castella?; Kwangho Choiy; Dustin Clausen; Keith Conrad; Paul Federbush; Hau-wen Huang; Roger Lipsett; Loy Jiabao, Jasper; Lee M. Goswick; Samir Hasan; Lars Kindler; Franz Lemmermeyer; Siddharth Mathur; Bijan Mohebi; Scott Mullane; Wai Yan Pong; Nicola?s Sirolli; Thomas Stoll; Vishne Uzi; and others. PARI is an open source computer algebra system freely available from http://pari.math.u- bordeaux.fr/. FERMAT (16011665). Stated his last theorem, and proved it for mD 4. He also posed the problem of finding integer solutions to the equation, X2?AY 2 D 1; A 2 Z; (1) which is essentially the problem1 of finding the units in Z? p A?.

The English mathemati- cians found an algorithm for solving the problem, but neglected to hot teen smoking prove that the algorithm always works. EULER (17071783). He introduced analysis into the study of the prime numbers, and he discovered an early version of the quadratic reciprocity law. About Effects! LAGRANGE (17361813). He found the complete form of the quadratic reciprocity law: D .?1/.p?1/.q?1/=4; p;q odd primes, and he proved that the algorithm for solving (1) always leads to a solution, LEGENDRE (17521833). He introduced the Legendre symbol m p. , and gave an incom- plete proof of the quadratic reciprocity law. He proved the following local-global principle for quadratic forms in three variables over Q: a quadratic form Q.X;Y;Z/ has a nontrivial zero in smoking Q if and only if it has one in R and the congruence Q 0 mod pn has a nontrivial solution for all p and n. GAUSS (17771855). He found the dalda ghee, first complete proofs of the quadratic reciprocity law. He studied the hot teen smoking, Gaussian integers Z?i ? in order to find a quartic reciprocity law. He studied the classification of binary quadratic forms over Z, which is closely related to the problem of finding the class numbers of quadratic fields.

DIRICHLET (18051859). He introduced L-series, and used them to prove an analytic for- mula for the class number and a density theorem for the primes in dalda ghee an arithmetic progression. He proved the following unit theorem: let ? be a root of a monic irreducible polynomial f .X/ with integer coefficients; suppose that f .X/ has r real roots and 2s complex roots; then Z??? is a finitely generated group of rank rC s?1. KUMMER (18101893). He made a deep study of the hot teen, arithmetic of Essay Phobias: Effects, cyclotomic fields, mo- tivated by a search for higher reciprocity laws, and showed that unique factorization could be recovered by the introduction of ideal numbers. He proved that Fermats last theorem holds for regular primes.

HERMITE (18221901). He made important contributions to quadratic forms, and he showed that the roots of smoking, a polynomial of degree 5 can be expressed in terms of elliptic functions. EISENSTEIN (18231852). He published the about Phobias: Effects and Treatments, first complete proofs for the cubic and quartic reciprocity laws. KRONECKER (18231891). He developed an alternative to Dedekinds ideals. He also had one of the most beautiful ideas in mathematics for generating abelian extensions of number fields (the Kronecker liebster Jugendtraum).

RIEMANN (18261866). Studied the hot teen, Riemann zeta function, and made the Riemann hy- pothesis. 1The Indian mathematician Bhaskara (12th century) knew general rules for finding solutions to the equa- tion. DEDEKIND (18311916). He laid the modern foundations of algebraic number theory by finding the correct definition of the ring of integers in a number field, by proving that ideals factor uniquely into 7 years in tibet true story, products of prime ideals in such rings, and by showing that, modulo principal ideals, they fall into finitely many classes. Defined the zeta function of a number field. WEBER (18421913). Made important progress in class field theory and the Kronecker Jugendtraum. HENSEL (18611941).

He gave the first definition of the field of p-adic numbers (as the set of infinite sums. n, an 2 f0;1; : : : ;p?1g). HILBERT (18621943). He wrote a very influential book on algebraic number theory in 1897, which gave the first systematic account of the theory. Some of his famous problems were on number theory, and have also been influential. TAKAGI (18751960). He proved the fundamental theorems of abelian class field theory, as conjectured by Weber and Hilbert. NOETHER (18821935). Together with Artin, she laid the foundations of modern algebra in hot teen smoking which axioms and conceptual arguments are emphasized, and she contributed to the classification of central simple algebras over number fields. HECKE (18871947).

Introduced HeckeL-series generalizing both DirichletsL-series and Dedekinds zeta functions. ARTIN (18981962). He found the Artin reciprocity law, which is the main theorem of class field theory (improvement of Takagis results). True Story! Introduced the Artin L-series. HASSE (18981979).

He gave the first proof of local class field theory, proved the Hasse (local-global) principle for all quadratic forms over number fields, and contributed to the classification of central simple algebras over number fields. BRAUER (19011977). Defined the Brauer group, and contributed to the classification of central simple algebras over number fields. WEIL (19061998). Defined the hot teen smoking, Weil group, which enabled him to Placed Women in Antigone and A House give a common gener- alization of Artin L-series and Hecke L-series. CHEVALLEY (190984). The main statements of hot teen smoking, class field theory are purely algebraic, but all the earlier proofs used analysis; Chevalley gave a purely algebraic proof. With his introduction of ide?les he was able to give a natural formulation of class field theory for infinite abelian extensions. IWASAWA (19171998). He introduced an important new approach into marijuanas should essay, algebraic number theory which was suggested by the theory of curves over finite fields.

TATE (1925 ). He proved new results in group cohomology, which allowed him to give an elegant reformulation of class field theory. With Lubin he found an explicit way of generating abelian extensions of local fields. LANGLANDS (1936 ). The Langlands program2 is a vast series of conjectures that, among other things, contains a nonabelian class field theory. 2Not to be confused with its geometric analogue, sometimes referred to as the geometric Langlands pro- gram, which appears to lack arithmetic significance. Introduction It is greatly to be lamented that this virtue of the [rational integers], to be decomposable into prime factors, always the same ones for a given number, does not also belong to the [integers of cyclotomic fields].

Kummer 1844 (as translated by Andre? Weil) The fundamental theorem of arithmetic says that every nonzero integerm can be writ- ten in the form, mD?p1 pn; pi a prime number, and that this factorization is essentially unique. Consider more generally an integral domain A. An element a 2A is said to be a unit if. it has an inverse in A (element b such that ab D 1D ba). I write A for the multiplicative group of units in A. An element of A is said to prime if it is neither zero nor a unit, and if. If A is a principal ideal domain, then every nonzero element a of A can be written in the form, aD u1 n; u a unit; i a prime element; and this factorization is unique up to order and replacing each i with an associate, i.e., with its product with a unit. Our first task will be to discover to what extent unique factorization holds, or fails to hold, in number fields. Three problems present themselves.

First, factorization in a field only makes sense with respect to a subring, and hot teen so we must define the ring of integers OK in our number field K. Secondly, since unique factorization will fail in general, we shall need to find a way of measuring by how much it fails. Finally, since factorization is only considered up to units, in order to fully understand the arithmetic of K, we need to why medical should be legal essay understand the structure of the group of units UK in OK . THE RING OF INTEGERS. Let K be an algebraic number field. Each element ? of K satisfies an equation. ?nCa1? n?1 C Ca0 D 0. with coefficients a1; : : : ;an in Q, and ? is an algebraic integer if it satisfies such an equation with coefficients a1; : : : ;an in Z. We shall see that the algebraic integers form a subring OK of K. The criterion as stated is difficult to apply. We shall show (2.11) that ? is an algebraic integer if and hot teen only if its minimum polynomial over Q has coefficients in Z. Consider for example the field K D Q? p d?, where d is a square-free integer. The. minimum polynomial of ? D aCb p d , b 0, a;b 2Q, is. .X ? .aCb p d//.X ? .a?b. p d//DX2?2aXC .a2?b2d/; and why medical should so ? is an algebraic integer if and hot teen smoking only if. 2a 2 Z; a2?b2d 2 Z: From this it follows easily that, when d 2;3 mod 4, ? is an algebraic integer if and only if a and b are integers, i.e., and, when d 1 mod 4, ? is an algebraic integer if and only if a and b are either both integers or both half-integers, i.e., For example, the minimum polynomial of 1=2C p 5=2 is X2?X ?1, and why medical marijuanas should essay so 1=2C. is an algebraic integer in Q? p 5?. Let d be a primitive d th root of 1, for example, d D exp.2i=d/, and letK DQ?d ?. Then we shall see (6.2) that. OK D Z?d ?D ?P.

as one would hope. A nonzero element of an integral domain A is said to be irreducible if it is not a unit, and cant be written as a product of two nonunits. For example, a prime element is (obviously) irreducible. A ring A is a unique factorization domain if every nonzero element of A can be expressed as a product of irreducible elements in essentially one way. Is the ring of integers OK a unique factorization domain?

No, not in general! We shall see that each element of OK can be written as a product of irreducible elements (this is true for all Noetherian rings), and so it is the uniqueness that fails. For example, in Z? p ?5? we have. 6D 2 3D .1C p ?5/.1?. To see that 2, 3, 1C p ?5, 1?. p ?5 are irreducible, and smoking no two are associates, we use the. p ?5 7! a2C5b2: This is multiplicative, and it is easy to see that, for ? 2OK , Nm.?/D 1 ? N? D 1 ? is a unit. (*) If 1C p ?5D ??, then Nm.??/D Nm.1C. p ?5/D 6. Thus Nm.?/D 1;2;3, or 6. In the. first case, ? is a unit, the second and third cases dont occur, and in the fourth case ? is a unit. A similar argument shows that 2;3, and 1?. Essay About Effects Of Phobias! p ?5 are irreducible. Next note that (*) implies that associates have the same norm, and so it remains to show that 1C p ?5 and.

1? p ?5 are not associates, but. has no solution with a;b 2 Z. Why does unique factorization fail in hot teen OK? The problem is that irreducible elements in. OK need not be prime. In the above example, 1C p ?5 divides 2 3 but it divides neither 2. nor 3. In fact, in an integral domain in which factorizations exist (e.g. a Noetherian ring), factorization is unique if all irreducible elements are prime. What can we recover? Consider. 210D 6 35D 10 21: If we were naive, we might say this shows factorization is not unique in of an New World Z; instead, we recognize that there is a unique factorization underlying these two decompositions, namely, The idea of Kummer and Dedekind was to enlarge the set of prime numbers so that, for smoking, example, in Z? p ?5? there is a unique factorization, 6D .p1 p2/.p3 p4/D .p1 p3/.p2 p4/; underlying the why medical, above factorization; here the pi are ideal prime factors.

How do we define ideal factors? Clearly, an ideal factor should be characterized. by the algebraic integers it divides. Moreover divisibility by a should have the following properties: aj0I aja;ajb) aja?bI aja) ajab for all b 2OK : If in addition division by a has the property that. ajab) aja or ajb; then we call a a prime ideal factor. Since all we know about an ideal factor is the smoking, set of elements it divides, we may as well identify it with this set. 7 Years Story! Thus an ideal factor a is a set of elements of OK such that. 0 2 aI a;b 2 a) a?b 2 aI a 2 a) ab 2 a for all b 2OK I. it is prime if an addition, ab 2 a) a 2 a or b 2 a: Many of you will recognize that an ideal factor is what we now call an ideal, and hot teen smoking a prime ideal factor is a prime ideal. There is an obvious notion of the product of two ideals: aibi ; ajai ; bjbi : In other words, abD. nX aibi j ai 2 a; bi 2 b. One see easily that this is again an ideal, and that if. aD .a1; . ;am/ and bD .b1; . ;bn/ then a bD .a1b1; . ;aibj ; . ;ambn/: With these definitions, one recovers unique factorization: if a 0, then there is an essentially unique factorization: .a/D p1 pn with each pi a prime ideal. In the above example, .6/D .2;1C p ?5/.2;1?.

In fact, I claim. .2;1C p ?5/.2;1?. .3;1C p ?5/.3;1?. .2;1? p ?5/.3;1?. For example, .2;1C p ?5/.2;1?. p ?5;6/. Since every gen- erator is divisible by 2, we see that. .2;1C p ?5/.2;1?. Conversely, 2D 6?4 2 .4;2C2. and so .2;1C p ?5/.2;1?. p ?5/ D .2/, as claimed. I further claim that the four ideals. .2;1C p ?5/, .2;1?. p ?5/, and .3;1?. p ?5/ are all prime.

For example, the obvious map Z! Z? p ?5?=.3;1?. p ?5/ is surjective with kernel .3/, and so. Z? p ?5?=.3;1?. which is an integral domain. How far is this from what we want, namely, unique factorization of elements? In other. words, how many ideal elements have we had to add to our real elements to get unique factorization. In a certain sense, only a finite number: we shall see that there exists a finite set S of ideals such that every ideal is of the form a .a/ for some a 2 S and some a 2OK . Better, we shall construct a group I of fractional ideals in 7 years true story which the principal fractional ideals .a/, a 2K, form a subgroup P of finite index.

The index is called the class number hK of K. We shall see that. hK D 1 OK is a principal ideal domain OK is a unique factorization domain. Unlike Z, OK can have infinitely many units. For example, .1C p 2/ is a unit of infinite. order in Z? p 2? W. p 2/m 1 if m 0: In fact Z? p 2? D f?.1C. p 2/m jm 2 Zg, and so. Smoking! Z? p 2? f?1gffree abelian group of rank 1g: In general, we shall show (unit theorem) that the roots of 1 in robinson K form a finite group .K/, and that. OK .K/Z r (as an abelian group); moreover, we shall find r: One motivation for smoking, the development of algebraic number theory was the attempt to prove Fermats last theorem, i.e., when m 3, there are no integer solutions .x;y;z/ to the equation. with all of x;y;z nonzero. WhenmD 3, this can proved by the method of infinite descent, i.e., from Essay of an Outsider’s in Brave one solution, you show that you can construct a smaller solution, which leads to a contradiction3. The proof makes use of the smoking, factorization. Y 3 DZ3?X3 D .Z?X/.Z2CXZCX2/; and it was recognized that a stumbling block to proving the theorem for larger m is that no such factorization exists into polynomials with integer coefficients of degree 2. This led people to look at more general factorizations. In a famous incident, the French mathematician Lame? gave a talk at the Paris Academy in 1847 in 7 years in tibet which he claimed to prove Fermats last theorem using the following ideas. Let p 2 be a prime, and suppose x, y, z are nonzero integers such that.

Write xp D zp?yp D. Y .z? iy/; 0 i p?1; D e2i=p: He then showed how to obtain a smaller solution to the equation, and hence a contradiction. Liouville immediately questioned a step in Lame?s proof in which he assumed that, in order to show that each factor .z ? iy/ is a pth power, it suffices to show that the factors are relatively prime in pairs and their product is a pth power. In fact, Lame? couldnt justify his step (Z?? is not always a principal ideal domain), and Fermats last theorem was not proved for almost 150 years. However, shortly after Lame?s embarrassing lecture, Kummer used his results on the arithmetic of the fields Q?? to prove Fermats last theorem for hot teen smoking, all regular primes, i.e., for all primes p such that p does not divide the class number of Q?p?. Another application is to finding Galois groups. The splitting field of a polynomial f .X/ 2Q?X? is a Galois extension of 7 years, Q. In a basic Galois theory course, we learn how to compute the Galois group only when the degree is very small. By using algebraic number theory one can write down an algorithm to do it for any degree. For applications of algebraic number theory to elliptic curves, see, for example, Milne 2006. Some comments on hot teen the literature. COMPUTATIONAL NUMBER THEORY.

Cohen 1993 and Phobias: and Treatments of Phobias Pohst and Zassenhaus 1989 provide algorithms for most of the construc- tions we make in this course. The first assumes the reader knows number theory, whereas the second develops the whole subject algorithmically. Cohens book is the more useful as a supplement to this course, but wasnt available when these notes were first written. While the books are concerned with more-or-less practical algorithms for fields of small degree and small discriminant, Lenstra (1992) concentrates on finding good general algorithms. 3The simplest proof by infinite descent is that showing that p 2 is irrational. HISTORY OF ALGEBRAIC NUMBER THEORY.

Dedekind 1996, with its introduction by Stillwell, gives an excellent idea of how algebraic number theory developed. Edwards 1977 is a history of algebraic number theory, con- centrating on the efforts to prove Fermats last theorem. The notes in Narkiewicz 1990 document the origins of most significant results in algebraic number theory. Lemmermeyer 2009, which explains the hot teen, origins of ideal numbers, and other writings by the same author, e.g., Lemmermeyer 2000, 2007. 0-1 Let d be a square-free integer. Complete the verification that the ring of integers in Q? p d? is as described. 0-2 Complete the verification that, in Z? p ?5?, .6/D .2;1C p ?5/.2;1?. is a factorization of .6/ into a product of prime ideals. CHAPTER 1 Preliminaries from Commutative.

Many results that were first proved for rings of integers in Women in Antigone Doll's number fields are true for more general commutative rings, and hot teen smoking it is more natural to prove them in that context.1. All rings will be commutative, and have an identity element (i.e., an element 1 such that 1a D a for all a 2 A), and a homomorphism of rings will map the identity element to the identity element. A ring B together with a homomorphism of rings A! B will be referred to as an A-algebra. Why Medical Marijuanas Should! We use this terminology mainly when A is a subring of B . In this case, for elements ?1; . ;?m of B , A??1; . ;?m? denotes the smallest subring of B containing A and the ?i . It consists of all polynomials in the ?i with coefficients in A, i.e., elements of the form X. ai1. im? i1 1 . ? im m ; ai1. im 2 A: We also refer to A??1; . Hot Teen! ;?m? as the A-subalgebra of B generated by 7 years in tibet true the ?i , and when B D A??1; . ;?m? we say that the hot teen smoking, ?i generate B as an A-algebra. For elements a1;a2; : : : of A, we let .a1;a2; : : :/ denote the smallest ideal containing the ai . It consists of finite sums. P ciai , ci 2 A, and why medical be legal it is called the ideal generated by. a1;a2; : : :. When a and b are ideals in A, we define. aCbD faCb j a 2 a, b 2 bg: It is again an ideal in A in fact, it is the smallest ideal containing both a and b. If aD .a1; . ;am/ and bD .b1; . ;bn/, then aCbD .a1; . ;am;b1; . ;bn/: Given an ideal a in A, we can form the quotient ring A=a. Let f WA!

A=a be the homomorphism a 7! aCa; then b 7! f ?1.b/ defines a one-to-one correspondence between the ideals of A=a and hot teen the ideals of A containing a, and. 1See also the notes A Primer of Commutative Algebra available on my website. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. A proper ideal a of Essay about Effects, A is prime if ab 2 a) a or b 2 a. An ideal a is prime if and only if the quotient ring A=a is an integral domain. A nonzero element of A is said to hot teen be prime if ./ is a prime ideal; equivalently, if jab) ja or jb. An ideal m in A is maximal if it is maximal among the proper ideals of A, i.e., if mA and there does not exist an ideal a A containing m but distinct from it. An ideal a is maximal if and only if A=a is a field. Every proper ideal a of A is contained in a maximal ideal if A is Noetherian (see below) this is obvious; otherwise the proof requires Zorns lemma.

In particular, every nonunit in A is contained in in tibet true story a maximal ideal. There are the implications: A is a Euclidean domain) A is a principal ideal domain ) A is a unique factorization domain (see any good graduate algebra course). Ideals in hot teen products of rings. PROPOSITION 1.1 Consider a product of rings AB . If a and b are ideals in A and B respectively, then ab is an ideal in AB , and Placed Upon in Antigone and A Essay every ideal in AB is hot teen smoking, of this form. The prime ideals of AB are the ideals of the form.

pB (p a prime ideal of A), Ap (p a prime ideal of B). Dalda Ghee! PROOF. Let c be an ideal in AB , and let. Smoking! aD fa 2 A j .a;0/ 2 cg; bD fb 2 B j .0;b/ 2 cg: Clearly a b c. Conversely, let .a;b/ 2 c. Then .a;0/ D .a;b/ .1;0/ 2 c and .0;b/ D .a;b/ .0;1/ 2 c, and so .a;b/ 2 ab: Recall that an ideal c C is prime if and only if C=c is an integral domain. The map. has kernel ab, and hence induces an isomorphism. Now use that a product of rings is an integral domain if and only if one ring is zero and the other is an integral domain. 2. REMARK 1.2 The lemma extends in an obvious way to a finite product of rings: the ideals in A1 Am are of the form a1 am with ai an ideal in Ai ; moreover, a1 am is dalda ghee, prime if and only if there is a j such that aj is a prime ideal in Aj and ai DAi for i j: A ring A is Noetherian if every ideal in A is finitely generated. PROPOSITION 1.3 The following conditions on a ring A are equivalent: (a) A is Noetherian. (b) Every ascending chain of ideals. eventually becomes constant, i.e., for some n, an D anC1 D . (c) Every nonempty set S of ideals in A has a maximal element, i.e., there exists an ideal in S not properly contained in any other ideal in S . PROOF. (a) (b): Let a D S. Hot Teen Smoking! ai ; it is an ideal, and hence is finitely generated, say a D .a1; : : : ;ar/. For some n, an will contain all the ai , and so an D anC1 D D a. (b) (c): Let a1 2 S . If a1 is not a maximal element of S , then there exists an a2 2 S such that a1 a2. If a2 is not maximal, then there exists an a3 etc.. From (b) we know that this process will lead to a maximal element after only finitely many steps. Why Medical Marijuanas Essay! (c) (a): Let a be an ideal in A, and let S be the set of finitely generated ideals contained in smoking a. Then S is nonempty because it contains the zero ideal, and so it contains a maximal element, say, a0 D .a1; : : : ;ar/.

If a0 a, then there exists an element a 2 ar a0, and .a1; : : : ;ar ;a/ will be a finitely generated ideal in a properly containing a0. This contradicts the definition of a0. Placed Antigone! 2. A famous theorem of Hilbert states that k?X1; . ;Xn? is Noetherian. In practice, al- most all the rings that arise naturally in algebraic number theory or algebraic geometry are Noetherian, but not all rings are Noetherian. For example, the hot teen smoking, ring k?X1; : : : ;Xn; : : :? of polynomials in an infinite sequence of symbols is not Noetherian because the chain of ideals. never becomes constant. PROPOSITION 1.4 Every nonzero nonunit element of a Noetherian integral domain can be written as a product of irreducible elements.

PROOF. We shall need to use that, for elements a and b of an integral domain A, .a/ .b/ bja, with equality if and friday crusoe only if b D aunit: The first assertion is obvious. For the second, note that if a D bc and b D ad then a D bc D adc, and so dc D 1. Hence both c and d are units. Suppose the statement of the proposition is false for a Noetherian integral domain A. Then there exists an element a 2 A which contradicts the hot teen smoking, statement and is such that .a/ is maximal among the ideals generated by such elements (here we use that A is Noetherian). Since a can not be written as a product of irreducible elements, it is not itself irreducible, and so a D bc with b and c nonunits.

Clearly .b/ .a/, and the ideals cant be equal for otherwise c would be a unit. From the maximality of .a/, we deduce that b can be written as a product of irreducible elements, and similarly for c. Thus a is a product of irreducible elements, and we have a contradiction. 2. REMARK 1.5 Note that the proposition fails for the ring O of all algebraic integers in the algebraic closure of Q in C, because, for example, we can keep in extracting square roots an algebraic integer ? can not be an irreducible element of O because. p ? will also be. an algebraic integer and ? D p ? p ?. Friday Robinson! Thus O is not Noetherian. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Let A be a ring. Hot Teen! An A-module M is said to be Noetherian if every submodule is finitely generated.

PROPOSITION 1.6 The following conditions on an A-module M are equivalent: (a) M is Noetherian; (b) every ascending chain of submodules eventually becomes constant; (c) every nonempty set of submodules in M has a maximal element. PROOF. About Effects Of Phobias! Similar to the proof of Proposition 1.3. 2. PROPOSITION 1.7 Let M be an A-module, and let N be a submodule of M . If N and M=N are both Noetherian, then so also is M . PROOF. I claim that if M 0 M 00 are submodules of M such that M 0N DM 00N and M 0 and M 00 have the same image in M=N , then M 0 DM 00. To see this, let x 2M 00; the second condition implies that there exists a y 2M 0 with the same image as x inM=N , i.e., such that x?y 2N . Then x?y 2M 00N M 0, and so x 2M 0. Now consider an hot teen smoking, ascending chain of submodules of M . If M=N is Noetherian, the image of the chain in M=N becomes constant, and Essay Perspective in Brave New World if N is Noetherian, the intersection of the chain with N becomes constant. Now the smoking, claim shows that the chain itself becomes constant. 2. PROPOSITION 1.8 Let A be a Noetherian ring.

Then every finitely generated A-module is Noetherian. PROOF. If M is generated by a single element, then M A=a for some ideal a in A, and the statement is obvious. We argue by induction on the minimum number n of generators ofM . SinceM contains a submoduleN generated by n?1 elements such that the quotient M=N is generated by a single element, the statement follows from (1.7). 2. A ring A is said to local if it has exactly one maximal ideal m. In this case, A D Arm (complement of m in A). LEMMA 1.9 (NAKAYAMAS LEMMA) Let A be a local Noetherian ring, and let a be a proper ideal in A. Let M be a finitely generated A-module, and define. aM D f P aimi j ai 2 a; mi 2M g : (a) If aM DM , then M D 0: (b) If N is a submodule of M such that N CaM DM , then N DM: Rings of fractions. PROOF. (a) Suppose that aM D M but M 0. Choose a minimal set of generators fe1; : : : ; eng for M , n 1, and write. e1 D a1e1C Canen, ai 2 a: Then .1?a1/e1 D a2e2C Canen: As 1? a1 is not in m, it is a unit, and so fe2; . ; eng generates M , which contradicts our choice of fe1; : : : ; eng. (b) It suffices to show that a.M=N/DM=N for then (a) shows that M=N D 0. Con- sider mCN , m 2M . From the assumption, we can write. aimi , with ai 2 a, mi 2M: and so mCN 2 a.M=N/: 2. The hypothesis that M be finitely generated in the lemma is essential. For example, if A is a local integral domain with maximal ideal m 0, then mM DM for any field M containing A but M 0. Rings of fractions.

Let A be an integral domain; there is a field K A, called the field of fractions of A, with the property that every c 2K can be written in 7 years in tibet the form c D ab?1 with a;b 2A and b 0. Hot Teen Smoking! For example, Q is the field of fractions of Z, and k.X/ is the field of fractions of k?X?: Let A be an integral domain with field of fractions K. Essay Effects! A subset S of A is said to be multiplicative if 0 S , 1 2 S , and S is closed under multiplication. If S is a multiplicative subset, then we define. S?1AD fa=b 2K j b 2 Sg: It is obviously a subring of K: EXAMPLE 1.10 (a) Let t be a nonzero element of smoking, A; then. St def D f1,t ,t2. g. is a multiplicative subset of A, and we (sometimes) write At for S?1t A. For example, if d is a nonzero integer, then2 Zd consists of those elements of Q whose denominator divides some power of d : Zd D fa=dn 2Q j a 2 Z, n 0g: (b) If p is a prime ideal, then SpDArp is a multiplicative set (if neither a nor b belongs to Essay of an New World p, then ab does not belong to p/. We write Ap for S?1p A. For example, Z.p/ D fm=n 2Q j n is not divisible by pg:

2This notation conflicts with a later notation in which Zp denotes the ring of hot teen smoking, p-adic integers. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. PROPOSITION 1.11 Consider an integral domainA and a multiplicative subset S ofA. For an ideal a of A, write ae for the ideal it generates in S?1A; for an ideal a of S?1A, write ac for aA. Then: ace D a for all ideals a of about Phobias:, S?1A aec D a if a is a prime ideal of hot teen smoking, A disjoint from S: PROOF. Let a be an ideal in S?1A. Clearly .aA/e a because aA a and a is an why medical essay, ideal in S?1A. For the hot teen, reverse inclusion, let b 2 a. We can write it b D a=s with a 2 A, s 2 S . Then aD s .a=s/ 2 aA, and so a=s D .s .a=s//=s 2 .aA/e: Let p be a prime ideal disjoint from S . Clearly .S?1p/A p. For the reverse inclu- sion, let a=s 2 .S?1p/A, a 2 p, s 2 S . Consider the equation a. s s D a 2 p. Both a=s. and s are in A, and so at of an Perspective New World, least one of a=s or s is in p (because it is prime); but s p (by assumption), and so a=s 2 p: 2. PROPOSITION 1.12 Let A be an integral domain, and let S be a multiplicative subset of A. The map p 7! pe defD p S?1A is a bijection from the set of smoking, prime ideals in A such that pS D? to the set of prime ideals in S?1A; the inverse map is p 7! pA. PROOF. It is easy to see that. p a prime ideal disjoint from S) pe is a prime ideal in 7 years in tibet true story S?1A, p a prime ideal in S?1A) pA is a prime ideal in A disjoint from S; and (1.11) shows that the two maps are inverse.

2. EXAMPLE 1.13 (a) If p is a prime ideal in A, then Ap is a local ring (because p contains every prime ideal disjoint from Sp). (b) We list the prime ideals in some rings: Note that in hot teen smoking general, for t a nonzero element of an integral domain, fprime ideals of Atg $ fprime ideals of A not containing tg. fprime ideals of A=.t/g $ fprime ideals of 7 years in tibet, A containing tg: The Chinese remainder theorem. Recall the classical form of the theorem: let d1; . ;dn be integers, relatively prime in pairs; then for any integers x1; . ;xn, the congruences. Hot Teen Smoking! The Chinese remainder theorem. have a simultaneous solution x 2 Z; moreover, if x is one solution, then the Placed Upon Women, other solutions are the integers of the form xCmd with m 2 Z and d D. We want to translate this in terms of ideals. Integersm and n are relatively prime if and only if .m;n/D Z, i.e., if and only if .m/C .n/D Z. This suggests defining ideals a and b in a ring A to be relatively prime if aCbD A. Hot Teen Smoking! If m1; . ;mk are integers, then T .mi / D .m/ where m is the least common multiple. of the mi . Thus T .mi / . Q mi /, which equals. Q .mi /. If the mi are relatively prime in. Friday Crusoe! pairs, then mD Q mi , and so we have. Q .mi /. Note that in general, a1 a2 an a1a2 . an; but the two ideals need not be equal. These remarks suggest the smoking, following statement. THEOREM 1.14 Let a1; . ;an be ideals in a ring A, relatively prime in pairs. Then for any elements x1; . ;xn of A, the congruences. have a simultaneous solution x 2 A; moreover, if x is one solution, then the other solutions are the elements of the form xC a with a 2. Q ai . In other words, the. natural maps give an exact sequence.

PROOF. Suppose first that n D 2. In Tibet! As a1C a2 D A, there are elements ai 2 ai such that a1Ca2 D 1. The element x D a1x2Ca2x1 has the required property. For each i we can find elements ai 2 a1 and bi 2 ai such that. ai Cbi D 1, all i 2: The product Q i2.ai Cbi /D 1, and lies in smoking a1C. Q i2 ai , and so. We can now apply the theorem in the case nD 2 to obtain an element y1 of A such that. y1 1 mod a1; y1 0 mod Y. These conditions imply. y1 1 mod a1; y1 0 mod aj , all j 1: Similarly, there exist elements y2; . ;yn such that. yi 1 mod ai ; yi 0 mod aj for j i: The element x D P xiyi now satisfies the requirements. 1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. It remains to prove that T. ai . We have already noted that T. ai . First suppose that nD 2, and about let a1Ca2 D 1, as before. Hot Teen! For c 2 a1a2, we have. c D a1cCa2c 2 a1 a2. which proves that a1 a2 D a1a2.

We complete the proof by induction. This allows us to assume that. T i2 ai . We showed above that a1 and. Q i2 ai are relatively. prime, and so a1 . The theorem extends to A-modules. THEOREM 1.15 Let a1; . ;an be ideals in A, relatively prime in pairs, and let M be an A-module. There is an dalda ghee, exact sequence: This can be proved in the same way as Theorem 1.14, but I prefer to use tensor products, which I now review. Review of tensor products. Let M , N , and P be A-modules. A mapping f WM N ! P is said to be A-bilinear if. f .mCm0;n/D f .m;n/Cf .m0;n/ f .m;nCn0/D f .m;n/Cf .m;n0/ f .am;n/D af .m;n/D f .m;an/ 9=; all a 2 A; m;m0 2M; n;n0 2N: i.e., if it is linear in each variable. A pair .Q;f / consisting of an A-module Q and an A-bilinear map f WM N !Q is called the tensor product of M and N if any other A- bilinear map f 0WM N ! P factors uniquely into f 0 D ? ?f with ?WQ!

P A-linear. The tensor product exists, and is unique (up to a unique isomorphism making the obvious diagram commute). We denote it by M ?AN , and we write .m;n/ 7! m?n for f . The pair .M ?AN;.m;n/ 7!m?n/ is characterized by each of the smoking, following two conditions: (a) The mapM N !M ?AN is A-bilinear, and any other A-bilinear mapM N ! P is of the form .m;n/ 7! ?.m?n/ for a unique A-linear map ?WM ?AN ! P ; thus. BilinA.M N;P /D HomA.M ?AN;P /: (b) TheA-moduleM?AN has as generators them?n,m2M , n2N , and as relations. 9=; all a 2 A; m;m0 2M; n;n0 2N: Tensor products commute with direct sums: there is a canonical isomorphism. Review of tensor products.

It follows that if M and N are free A-modules3 with bases .ei / and .fj / respectively, then M ?AN is a free A-module with basis .ei ? fj /. In particular, if V and W are vector spaces over a field k of dimensions m and n respectively, then V ?kW is a vector space over k of dimension mn. Let ?WM !M 0 and Essay on Use Outsider’s Perspective New World ?WN !N 0 be A-linear maps. Then. .m;n/ 7! ?.m/??.n/WM N !M 0?AN 0. is A-bilinear, and therefore factors uniquely through M N !M ?AN . Thus there is a unique A-linear map ???WM ?AN !M 0?AN 0 such that. REMARK 1.16 The tensor product of two matrices regarded as linear maps is called their Kronecker product.4 If A is mn (so a linear map kn! km) and B is r s (so a linear map ks! kr ), then A?B is the mr ns matrix (linear map kns! kmr ) with. Hot Teen! 0B@ a11B a1nB. : : : . am1B amnB. 1CA : LEMMA 1.17 If ?WM !M 0 and ?WN !N 0 are surjective, then so also is. ???WM ?AN !M 0 ?AN. PROOF. On Use Of An Outsider’s New World! Recall that M 0?N 0 is generated as an A-module by the elements m0?n0, m0 2 M 0, n0 2 N 0. Smoking! By assumption m0 D ?.m/ for some m 2M and n0 D ?.n/ for some n 2 N , and som0?n0 D ?.m/??.n/D .???/.m?n/. Therefore the image of ??? contains a set of generators for M 0?AN 0 and so it is equal to it.

2. One can also show that if M 0!M !M 00! 0. is exact, then so also is. M 0?AP !M ?AP !M 00 ?AP ! 0: For example, if we tensor the exact sequence. with M , we obtain an exact sequence. a?AM !M ! .A=a/?AM ! 0 (2) 3Let M be an A-module. Elements e1; : : : ; em form a basis for M if every element of M can be expressed uniquely as a linear combination of the ei s with coefficients in A. Then Am!M , .a1; : : : ;am/ 7! an isomorphism of A-modules, and M is said to be a free A-module of rank m. 4Kronecker products of matrices pre-date tensor products by about 70 years.

1. PRELIMINARIES FROM COMMUTATIVE ALGEBRA. Dalda Ghee! The image of smoking, a?AM in M is. P aimi j ai 2 a, mi 2M g ; and so we obtain from the exact sequence (2) that. By way of contrast, ifM !N is injective, thenM ?AP !N ?AP need not be injective. For example, take A D Z, and note that .Z. m ! Z/?Z .Z=mZ/ equals Z=mZ. which is the zero map. PROOF (OF THEOREM 1.15) Return to the situation of the theorem. When we tensor the isomorphism. with M , we get an isomorphism. M=aM ' .A=a/?AM ' ! Q .A=ai /?AM ' EXTENSION OF SCALARS.

If A! B is an A-algebra and M is an Essay about Phobias: Effects of Phobias, A-module, then B?AM has a natural structure of smoking, a B-module for which. b.b0?m/D bb0?m; b;b0 2 B; m 2M: We say that B?AM is the B-module obtained from M by extension of scalars. The map m 7! 1?mWM ! B ?AM has the following universal property: it is A-linear, and for any A-linear map ?WM ! N from M into a B-module N , there is a unique B-linear map ?0WB?AM !N such that ?0.1?m/D ?.m/. Thus ? 7! ?0 defines an isomorphism. HomA.M;N /! HomB.B?AM;N/, N a B-module: For example, A?AM DM . If M is a free A-module with basis e1; : : : ; em, then B?AM is a free B-module with basis 1? e1; : : : ;1? em. TENSOR PRODUCTS OF ALGEBRAS.

If f WA! B and gWA! C are A-algebras, then B ?A C has a natural structure of an A-algebra: the product structure is determined by the rule. .b? c/.b0? c0/D bb0? cc0. and the map A! B?AC is a 7! f .a/?1D 1?g.a/. For example, there is a canonical isomorphism. a?f 7! af WK?k k?X1; : : : ;Xm?!K?X1; : : : ;Xm? (4) Review of tensor products. TENSOR PRODUCTS OF FIELDS. We are now able to compute K?k? if K is a finite separable field extension of a field k and ? is an arbitrary field extension of k. According to the primitive element theorem (FT 5.1), K D k??? for some ? 2K. Let f .X/ be the minimum polynomial of ?. By definition this means that the map g.X/ 7! g.?/ determines an isomorphism.

Hence K?k? ' .k?X?=.f .X///?k? '??X?=.f .X// by (3) and (4). Because K is dalda ghee, separable over k, f .X/ has distinct roots. Therefore f .X/ factors in ??X? into monic irreducible polynomials. that are relatively prime in pairs. We can apply the Chinese Remainder Theorem to deduce that. Finally, ??X?=.fi .X// is hot teen, a finite separable field extension of ? of degree degfi . Thus we have proved the following result: THEOREM 1.18 Let K be a finite separable field extension of k, and let ? be an arbitrary field extension. Then K?k? is a product of finite separable field extensions of ?, If ? is a primitive element for Upon Antigone, K=k, then the image ?i of ? in ?i is a primitive element for?i=?, and if f .X/ and fi .X/ are the minimum polynomials for ? and ?i respectively, then. EXAMPLE 1.19 Let K DQ??? with ? algebraic over Q. Then. C?QK ' C?Q .Q?X?=.f .X///' C?X?=..f .X//' Yr. iD1 C?X?=.X ??i / Cr : Here ?1; : : : ;?r are the conjugates of ? in C. The composite of ? 7! 1??WK!

C?QK with projection onto the i th factor is. We note that it is essential to assume in (1.18) that K is separable over k. If not, there will be an ? 2K such that ?p 2 k but ? k, and the ring K?kK will contain an element ? D .??1?1??/ 0 such that. ?p D ?p?1?1??p D ?p.1?1/??p.1?1/D 0: Hence K?kK contains a nonzero nilpotent element, and so it cant be a product of fields. NOTES Ideals were introduced and studied by Dedekind for smoking, rings of algebraic integers, and later by others in polynomial rings. It was not until the 1920s that the theory was placed in Outsider’s its most natural setting, that of arbitrary commutative rings (by Emil Artin and Emmy Noether). Embed this document on your website. If you don't receive any email, please check your Junk Mail box.

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