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Abstract Algebra | Group (Mathematics) | Field (Mathematics)

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Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University August 27, 2010 ii Copyright 1997 by Thomas W. Judson. Permission is granted to copy, distribute and/or modify this document un- der the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invari- ant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of
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  Abstract Algebra Theory and Applications Thomas W. JudsonStephen F. Austin State UniversityAugust 27, 2010  iiCopyright 1997 by Thomas W. Judson.Permission is granted to copy, distribute and/or modify this document un-der the terms of the GNU Free Documentation License, Version 1.2 or anylater version published by the Free Software Foundation; with no Invari-ant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the appendix entitled “GNU Free DocumentationLicense”.A current version can always be found via abstract.pugetsound.edu .  Preface This text is intended for a one- or two-semester undergraduate course inabstract algebra. Traditionally, these courses have covered the theoreti-cal aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract al-gebra and discrete mathematics have become increasingly important, andmany science, engineering, and computer science students are now electingto minor in mathematics. Though theory still occupies a central role in thesubject of abstract algebra and no student should go through such a coursewithout a good notion of what a proof is, the importance of applicationssuch as coding theory and cryptography has grown significantly.Until recently most abstract algebra texts included few if any applica-tions. However, one of the major problems in teaching an abstract algebracourse is that for many students it is their first encounter with an environ-ment that requires them to do rigorous proofs. Such students often find ithard to see the use of learning to prove theorems and propositions; appliedexamples help the instructor provide motivation.This text contains more material than can possibly be covered in a singlesemester. Certainly there is adequate material for a two-semester course,and perhaps more; however, for a one-semester course it would be quite easyto omit selected chapters and still have a useful text. The order of presen-tation of topics is standard: groups, then rings, and finally fields. Emphasiscan be placed either on theory or on applications. A typical one-semestercourse might cover groups and rings while briefly touching on field theory,using Chapters 1 through 6, 9, 10, 11, 13 (the first part), 16, 17, 18 (the firstpart), 20, and 21. Parts of these chapters could be deleted and applicationssubstituted according to the interests of the students and the instructor. Atwo-semester course emphasizing theory might cover Chapters 1 through 6,9, 10, 11, 13 through 18, 20, 21, 22 (the first part), and 23. On the otheriii  iv PREFACE  hand, if applications are to be emphasized, the course might cover Chapters1 through 14, and 16 through 22. In an applied course, some of the more the-oretical results could be assumed or omitted. A chapter dependency chartappears below. (A broken line indicates a partial dependency.)Chapter 23Chapter 22Chapter 21Chapter 18 Chapter 20 Chapter 19Chapter 17 Chapter 15Chapter 13 Chapter 16 Chapter 12 Chapter 14Chapter 11Chapter 10Chapter 8 Chapter 9 Chapter 7Chapters 1–6Though there are no specific prerequisites for a course in abstract alge-bra, students who have had other higher-level courses in mathematics willgenerally be more prepared than those who have not, because they will pos-sess a bit more mathematical sophistication. Occasionally, we shall assumesome basic linear algebra; that is, we shall take for granted an elemen-tary knowledge of matrices and determinants. This should present no greatproblem, since most students taking a course in abstract algebra have beenintroduced to matrices and determinants elsewhere in their career, if theyhave not already taken a sophomore- or junior-level course in linear algebra.
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