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This text is intended for a one- or two-semester undergraduate course in abstract algebra.Though there are no specific prerequisites for a course in abstract algebra, Students who have had other higher-level courses in mathematics will generally be more prepared than those who have not, because they will possess a bit more mathematical sophistication. Occasionally, we shall assume some basic linear algebra; that is, we shall take for granted an elementary knowledge of matrices and determinants.

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Abstract Algebra
Theory and Applications
Thomas W. JudsonStephen F. Austin State UniversityMay 7, 2013
iiCopyright 1997 by Thomas W. Judson.
Permission is granted to copy, distribute and/or modify this document underthe terms of the GNU Free Documentation License, Version 1.2 or any laterversion published by the Free Software Foundation; with no Invariant Sections,
no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is
included in the appendix entitled “GNU Free Documentation License”.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 theoreticalaspects of groups, rings, and ﬁelds. However, with the development of computing in the last several decades, applications that involve abstract
algebra and discrete mathematics have become increasingly important, and
many science, engineering, and computer science students are now electing
to minor in mathematics. Though theory still occupies a central role in thesubject of abstract algebra and no student should go through such a course
without a good notion of what a proof is, the importance of applications
such as coding theory and cryptography has grown signiﬁcantly.
Until recently most abstract algebra texts included few if any applications.
However, one of the major problems in teaching an abstract algebra course
is that for many students it is their ﬁrst encounter with an environment that
requires them to do rigorous proofs. Such students often ﬁnd it hard to see
the use of learning to prove theorems and propositions; applied examples
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 easy to
omit selected chapters and still have a useful text. The order of presentationof topics is standard: groups, then rings, and ﬁnally ﬁelds. Emphasis can be
placed either on theory or on applications. A typical one-semester coursemight cover groups and rings while brieﬂy touching on ﬁeld theory, usingChapters 1 through 6, 9, 10, 11, 13 (the ﬁrst part), 16, 17, 18 (the ﬁrst
part), 20, and 21. Parts of these chapters could be deleted and applications
substituted according to the interests of the students and the instructor. A
two-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 other
iii
iv
PREFACE
hand, if applications are to be emphasized, the course might cover Chapters
1 through 14, and 16 through 22. In an applied course, some of the more
theoretical results could be assumed or omitted. A chapter dependency chart
appears 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 1–6
Though there are no speciﬁc prerequisites for a course in abstract algebra,students who have had other higher-level courses in mathematics will generally
be more prepared than those who have not, because they will possess a bit
more mathematical sophistication. Occasionally, we shall assume some basic
linear algebra; that is, we shall take for granted an elementary knowledge
of matrices and determinants. This should present no great problem, since
most students taking a course in abstract algebra have been introduced to
matrices and determinants elsewhere in their career, if they have not already
taken a sophomore- or junior-level course in linear algebra.

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