# Fundamentals of Modern Algebra

## A Global Perspective

## Robert G Underwood

- 232 pages
- English
- ePUB (mobile friendly)
- Available on iOS & Android

# Fundamentals of Modern Algebra

## A Global Perspective

## Robert G Underwood

## About This Book

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

## Frequently asked questions

## Information

## Chapter 1

## Groups

*Z*we obtain greatest common divisors, least common multiples, Bezoutâs 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 Cauchyâs Theorem and Sylowâs First, Second, and Third Theorems.

### 1.1Introduction to Groups

*Z*, the multiplicative group of non-zero real numbers,

^{Ă}and the group of residue classes modulo

*n*,

*Z*. For further examples of groups we construct the 3rd and 4th dihedral groups,

_{n}*D*

_{3},

*D*

_{4}as the groups of symmetries of the equilateral triangle and the square, as well as the symmetric group on

*n*letters,

*S*.

_{n}*S*be a non-empty set of elements. The cartesian product on

*S*is defined as

*S*Ă

*S*= {(

*a*,

*b*) :

*a*,

*b*â

*S*}.

**Definition 1.1. A binary operation on**

*S*is a function

*B*:

*S*Ă

*S*â

*S*; we denote the image of (

*a*,

*b*) by

*ab*.

**commutative**if for...