Algebra

Larry C. Grove

• Publisher: Dover Publications
• Year: 2012
• eBook ISBN: 9780486142135
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

This graduate-level text is intended for initial courses in algebra that begin with first principles but proceed at a faster pace than undergraduate-level courses. It employs presentations and proofs that are accessible to students, and it provides numerous concrete examples.
Exercises appear throughout the text, clarifying concepts as they arise; additional exercises, varying widely in difficulty, are included at the ends of the chapters. Subjects include groups, rings, fields and Galois theory, modules, and structure of rings and algebras. Further topics encompass infinite Abelian groups, transcendental field extensions, representations and characters of finite groups, Galois groups, and additional areas.
Based on many years of classroom experience, this self-contained treatment breathes new life into abstract concepts.

Chapter I

Groups

1. GROUPS, SUBGROUPS, AND HOMOMORPHISMS

A nonempty set with an associative binary operation is called a semigroup, and a semigroup S having an identity element 1 such that 1x = x1 = x for all x ∈ S is called a monoid. Most of the algebraic systems discussed herein will be semigroups or monoids, but almost always with further requirements imposed, so the semigroup or monoid aspect will seldom be explicitly emphasized.

One trivial consequence of the definition of a monoid deserves mention.

Proposition 1.1. The identity element of a monoid S is unique.

Proof. Suppose 1 and e are identities in S. Then 1 = 1e = e.

A group is a set G with an associative binary operation (usually called multiplication) and an identity element 1 satisfying the further requirement that for each x ∈ G there is an inverse element y ∈ G such that xy = yx = 1.

Proposition 1.2. If G is a group and x ∈ G, then x has a unique inverse element.

Proof. Let y and z be inverses for x. Then

y = y1 = y(xz) = (yx)z = 1z = z.

The unique inverse for x ∈ G is denoted by x^-1. Note that (x^-1)^-1 = x.

Proposition 1.3. If G is a group and x, y ∈ G, then (xy)^-1 = y^-1 x^-1.

Proof

(xy)(y^-1 x^-1) = ((xy)y^-1)x^-1 = (x(yy^-1))x^-1 = (x1)x^-1 = xx^-1 = 1, and similarly (y^-1 x^-1)(xy) = 1.

As Coxeter [7] has pointed out, the “reversal of order” in Proposition 1.3 becomes clear when we think of the operations of putting on our shoes and socks.

If the binary operation of a group G is written as addition, then the identity element is commonly denoted by 0 rather than 1, and the i...

Table of contents

• Title Page
• Copyright Page
• Table of Contents
• Preface
• List of Symbols
• Introduction
• Errata
• Chapter I - Groups
• Chapter II - Rings
• Chapter III - Fields and Galois Theory
• Chapter IV - Modules
• Chapter V - Structure of Rings and Algebras
• Chapter VI - Further Topics
• Appendix - Zorn’s Lemma
• References
• Index