Abstract Algebra

Format:

Book

Author/Creator:

Judson, Thomas W., author.

Language:

English

Subjects (All):

Mathematics--Textbooks.

Mathematics.

Physical Description:

1 online resource

Place of Publication:

[Place of publication not identified] University of Puget Sound [2016]

Language Note:

In English.

Summary:

This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects 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 the subject 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 single semester. 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 presentation of topics is standard: groups, then rings, and finally fields. Emphasis can be placed either on theory or on applications. A typical one-semester course 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 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 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.)

Contents:

Preliminaries

The Integers

Groups

Cyclic Groups

Permutation Groups

Cosets and Lagrange's Theorem

Introduction to Cryptography

Algebraic Coding Theory

Isomorphisms

Normal Subgroups and Factor Groups

Homomorphisms

Matrix Groups and Symmetry

The Structure of Groups

Group Actions

The Sylow Theorems

Rings

Polynomials

Integral Domains

Lattices and Boolean Algebras

Vector Spaces

Fields

Finite Fields

Galois Theory

Notes:

Description based on print resource