Fundamentals of Modern Algebra

A Global Perspective

Robert G Underwood

• Verlag: WSPC
• Jahr: 2015
• ISBN:
  9789814730310
• Thema: Mathematics
• Thema: Discrete Mathematics

The purpose of this book is to provide a concise yet detailed account of fundamental concepts in modern algebra. The target audience for this book is first-year graduate students in mathematics, though the first two chapters are probably accessible to well-prepared undergraduates. The book covers a broad range of topics in modern algebra and includes chapters on groups, rings, modules, algebraic extension fields, and finite fields. Each chapter begins with an overview which provides a road map for the reader showing what material will be covered. At the end of each chapter we collect exercises which review and reinforce the material in the corresponding sections. These exercises range from straightforward applications of the material to problems designed to challenge the reader. We also include a list of "Questions for Further Study" which pose problems suitable for master's degree research projects.

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Contents:

• Groups
• Rings
• Modules
• Simple Algebraic Extension Fields
• Finite Fields

Readership: Graduate students in algebra.
Groups;Rings;Modules;Algebraic Extension Fields;Finite Fields Key Features:

• A broad range of essential topics in modern algebra are included in a relatively short book
• The fundamental results on the structure of groups, namely the structure theorem for finitely generated abelian groups, Cauchy's Theorem and Sylow's Theorems are included
• We include a detailed discussion of localizations and absolute values and completions. There is a natural progression from modules over fields (vector spaces) to modules over Noetherian rings. We also include a thorough discussion of the discriminant which leads to a concise yet readable introduction to algebraic number theory

Chapter 1

Groups

In this chapter we introduce semigroups, monoids, and groups, give some basic examples of groups and discuss some of their elementary properties. We then consider subgroups, cosets and Lagrange’s theorem, normal subgroups and the quotient group. We next turn to the basic maps between groups: homomorphisms and isomorphisms and their kernels. (Throughout this book, map = function.) We give the First, Second and Third Isomorphism theorems and the Universal Mapping Property for Kernels.

We close the chapter with the study of group structure, including generating sets for groups and subgroups and the notion of a cyclic group. From the cyclicity of the additive group of integers Z we obtain greatest common divisors, least common multiples, Bezout’s Lemma and the Chinese Remainder Theorem. We state the structure theorem for finitely generated abelian groups. Regarding the structure of groups in general, we introduce G-sets, and give Cauchy’s Theorem and Sylow’s First, Second, and Third Theorems.

1.1Introduction to Groups

In this section we define semigroups and monoids and give some examples, including the monoid of words on a finite alphabet. From semigroups and monoids, we develop the concept of a group, discuss finite, infinite and abelian groups, and prove some elementary properties of groups. We introduce examples of groups that we will appear throughout this book, including the additive group of integers, Z, the multiplicative group of non-zero real numbers,

^× and the group of residue classes modulo n, Z_n. For further examples of groups we construct the 3rd and 4th dihedral groups, D₃, D₄ as the groups of symmetries of the equilateral triangle and the square, as well as the symmetric group on n letters, S_n.

Let S be a non-empty set of elements. The cartesian product on S is defined as S × S = {(a, b) : a, b ∈ S}.

Definition 1.1. A binary operation on S is a function B : S × S → S; we denote the image of (a, b) by ab.

A binary operation is commutative if for...