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# Elements of Abstract Algebra

## Allan Clark

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- English
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eBook - ePub

# Elements of Abstract Algebra

## Allan Clark

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## About This Book

This concise, readable, college-level text treats basic abstract algebra in remarkable depth and detail. An antidote to the usual surveys of structure, the book presents group theory, Galois theory, and classical ideal theory in a framework emphasizing proof of important theorems.

Chapter I (Set Theory) covers the basics of sets. Chapter II (Group Theory) is a rigorous introduction to groups. It contains all the results needed for Galois theory as well as the Sylow theorems, the Jordan-Holder theorem, and a complete treatment of the simplicity of alternating groups. Chapter III (Field Theory) reviews linear algebra and introduces fields as a prelude to Galois theory. In addition there is a full discussion of the constructibility of regular polygons. Chapter IV (Galois Theory) gives a thorough treatment of this classical topic, including a detailed presentation of the solvability of equations in radicals that actually includes solutions of equations of degree 3 and 4 ― a feature omitted from all texts of the last 40 years. Chapter V (Ring Theory) contains basic information about rings and unique factorization to set the stage for classical ideal theory. Chapter VI (Classical Ideal Theory) ends with an elementary proof of the Fundamental Theorem of Algebraic Number Theory for the special case of Galois extensions of the rational field, a result which brings together all the major themes of the book.

The writing is clear and careful throughout, and includes many historical notes. Mathematical proof is emphasized. The text comprises 198 articles ranging in length from a paragraph to a page or two, pitched at a level that encourages careful reading. Most articles are accompanied by exercises, varying in level from the simple to the difficult.

Chapter I (Set Theory) covers the basics of sets. Chapter II (Group Theory) is a rigorous introduction to groups. It contains all the results needed for Galois theory as well as the Sylow theorems, the Jordan-Holder theorem, and a complete treatment of the simplicity of alternating groups. Chapter III (Field Theory) reviews linear algebra and introduces fields as a prelude to Galois theory. In addition there is a full discussion of the constructibility of regular polygons. Chapter IV (Galois Theory) gives a thorough treatment of this classical topic, including a detailed presentation of the solvability of equations in radicals that actually includes solutions of equations of degree 3 and 4 ― a feature omitted from all texts of the last 40 years. Chapter V (Ring Theory) contains basic information about rings and unique factorization to set the stage for classical ideal theory. Chapter VI (Classical Ideal Theory) ends with an elementary proof of the Fundamental Theorem of Algebraic Number Theory for the special case of Galois extensions of the rational field, a result which brings together all the major themes of the book.

The writing is clear and careful throughout, and includes many historical notes. Mathematical proof is emphasized. The text comprises 198 articles ranging in length from a paragraph to a page or two, pitched at a level that encourages careful reading. Most articles are accompanied by exercises, varying in level from the simple to the difficult.

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## Information

# Chapter 1

# Set Theory

Set theory is the proper framework for abstract mathematical thinking. All of the abstract entities we study in this book can be viewed as sets with specified additional structure. Set theory itself may be developed axiomatically, but the goal of this chapter is simply to provide sufficient familiarity with the notation and terminology of set theory to enable us to state definitions and theorems of abstract algebra in set-theoretic language. It is convenient to add some properties of the natural numbers to this informal study of set theory.

It is well known that an informal point of view in the theory of sets leads to contradictions. These difficulties all arise in operations with very large sets. We shall never need to deal with any sets large enough to cause trouble in this way, and, consequently, we may put aside all such worries.

# The Notation and Terminology of Set Theory

1. A set is any aggregation of objects, called elements of the set. Usually the elements of a set are mathematical quantities of a uniform character. For example, we shall have frequent occasion to consider the set of integers {. . . , −2, −1, 0, 1, 2, . . .}, which is customarily denoted Z (for the German “Zahlen,” which means “numbers”). We shall use also the set Q of rational numbers—numbers which are the quotient of two integers, such as 7/3, −4/5, 2.

To give an example of another type, we let K denote the set of coordinate points (x, y) in the xy-coordinate plane such that x

^{2}+ y^{2}= 1. Then K is the circle of unit radius with the origin as center.2. To indicate that a particular quantity x is an element of the set S, we write x ∈ S, and to indicate that it is not, we write x ∉ S. Thus −2 ∈ Z, but 1/2 ∉ Z; and 1/2 ∈ Q, but √2 ∉ Q.

A set is completely determined by its elements. Two sets are equal if and only if they have precisely the same elements. In oth...