An Invitation to Abstract Algebra

Steven J. Rosenberg

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781000516333
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics.

To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.

The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois' achievement in understanding how we can—and cannot—represent the roots of polynomials.

The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory.

The presentation includes the following features:

• Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters.

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• The text can be used for a one, two, or three-term course.

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• Each new topic is motivated with a question.

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• A collection of projects appears in Chapter 23.

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Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks—period. This book is offered as a manual to a new way of thinking. The author's aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

1 Review of Sets, Functions, and Proofs

DOI: 10.1201/9781003252139-1

1.1 Sets

Of all the things studied in mathematics, the most fundamental is the set. Although the notion of sets as the basis for thinking about math was only introduced relatively recently (for mathematics!) by Georg Cantor in the 1870s, sets now occupy a central position at the heart of mathematics.

A set is to be thought of as a collection of things, called the elements of the set. Elements of a set can be any type of thing: numbers, functions (to be defined later), or even other sets. Two sets are considered equal when they have precisely the same elements. We use the ordinary equality symbol to indicate that two sets are equal, as in “S = T.”

Notation 1.1. Let S be a set. The statement

$$\begin{array}{l}x\in S,\hfill \end{array}$$

read “x is in S” or “x in S,” means that x is an element of the set S. The statement

$$\begin{array}{l}T\subseteq S,\hfill \end{array}$$

read “T is a subset of S,” means that each element of T is also an element of S. The statement

$$\begin{array}{l}T\subset S,\hfill \end{array}$$

read “T is a proper subset of S,” means that T is a subset of S and $T\ne S$.

Definition 1.2. The cardinality of a set is the number of distinct elements in the set. We denote the cardinality of the set S by $\left|S\right|$. We also speak of cardinality as size.

Definition 1.3. A singleton set is a set of cardinality 1.

1.1.1 Some Special Sets of Numbers

We have special notation for certain of the most important sets of numbers:

Z is the set of all integers, or whole numbers; we use the notation Z⁺ to denote the set of all positive integers;

N is the set of all natural numbers, or non-negative integers (note that we include 0 as an element of N, but some other authors do not);

Q is the set of all rational numbers, i.e., numbers which are ratios of two integers, where the denominator is not zero;

R is the set of all real numbers; and finally,

C is the set of all complex numbers.

As the first and most basic set of numbers in our list above, we remind the reader of some of the properties of Z. Given two integers a and b, we say that b divides a, written , if there is some integer c such that $a=b·c$, where ⋅ is ordinary multiplication.

The greatest common divisor of a and b, written $\gcd (a,b)$, is the largest integer d such that both and ; the gcd exists if $a\ne 0$ or $b\ne 0$.

The least common multiple...