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Copyright © 2008, 1980 by Dan Saracino

I O-digit ISBN 1-57766-536-8 13-digit ISBN 978-1-57766-536-6

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CONTENTS

o Sets and Induction ............ ......... .... .... ................ ......... .... ........ ... ............... ...... 1 1 Binary Operations ..... .... ... ........... .. ... ......... ... .......... .... .. .. ....... ..... .. ... .. ........... 10 2 Groups .. ... ............. ..... .................. ... ........... ........ ............. ....... .... .... .. ... .......... 16 3 Fundamental Theorems about Groups ...... ....... .. ... .......... ...... ........ .......... ...... 25 4 Powers of an Element; Cyclic Groups .......................................................... 33 5 Subgroups .......... .......... ................ .. ........ ....... ........ ............... ... ....... ........ ........ 43 6 Direct Products ............................................................................................. 55 7 Functions ...................................................................................................... 59 8 Symmetric Groups ................................................ ..................... ........... ....... 66 9 Equivalence Relations; Cosets ....... ... .... ... ............... ............. ............... ......... 80

10 Counting the Elements of a Finite Group ..................................................... 88 11 Normal Subgroups ....................................................................................... 99 12 Homomorphisms ........................................................................................ 109 13 Homomorphisms and Normal Subgroups .................................... ..... .. ... .... 121 14 Direct Products and Finite Abelian Groups ....... ... .......... ......... .... ... ...... ...... 133 15 Sylow Theorems ........................................................................... ..... ... .. .... 143 16 Rings .......................................................................................................... 153 17 Subrings, Ideals, and Quotient Rings ....................... .... ............ .. ....... ......... 164 18 Ring Homomorphisms ............................................................................... 177 19 Polynomials .................. ..... ....... ..... .............. ............ ........... ................. .... ... 191 20 From Polynomials to Fields .... ......... ........ ...... .. ....... .................. ........... ... ... 205 21 Unique Factorization Domains ...... .... ..... ........ ............ .......... ......... ... .......... 211 22 Extensions of Fields ................. ..... .. .... ....................... ............... ........ ...... .. . 227 23 Constructions with Straightedge and Compass .......... .. ............ ........... ... .. .. 240 24 Normal and Separable Extensions ... ..... .... ....................... ......... ....... ..... ... ... 249 25 Galois Theory ............................................................................................. 265 26 Solvability ........... ... ........... ................... ...... .... ............................................ 279

Suggestions for Further Reading .... .. .......... ............... ............ ..... .... ........ ... 297 Answers to Selected Exercises ....................... ....... ........ ............. ... .... ......... 301 Index .......................................................................................... ........ ....... .. 307

PREFACE

This book is intended for use in a junior-senior level course in abstract algebra. The main change from the first edition is the addition of five new sections on field extensions and Galois theory, providing enough material for a two- semester course. More minor changes include the simplification of some points in the presentation, the addition of some new exercises, and the updating of some historical material.

In the earlier sections of the book I have preserved the emphasis on providing a large number of examples and on helping students learn how to write proofs. In the new sections the presentation is at a somewhat higher level. Unusual features, for a book that is still relatively short, are the inclusion of full proofs of both directions of Gauss' theorem on constructible regular polygons and Galois ' theorem on solvability by radicals, a Galois-theoretic proof of the Fundamental Theorem of Algebra, and a proof of the Primitive Element Theorem.

A one-semester course should probably include the material of Sections 0-13 , and some of the material on rings in Section 16 and the following sections. Sections 14 and 15 allow the inclusion of some deeper results on groups. The results of Section 14 are used in Section 15, and the First Sylow Theorem from Section 15 is used in Sections 25 and 26.

In two semesters it should be possible to cover the whole book, possibly omitting Section 21.

I want to express my appreciation to my students who used the manuscript for the five new sections as a text and pointed out to me parts of the presentation that needed clarification. I also want to thank all those who have sent me comments about the book over the years, and those who suggested that a new edition would be a good idea. I hope this second edition will be useful.

Dan Saracino

SECTION 0

SETS AND INDUCTION

One of the most fundamental notions in any part of mathematics is that of a set. You are probably already familiar with the basics about sets, but we will start out by running through them quickly, if for no other reason than to establish some notational conventions. After these generalities, we will make some remarks about the set of positive integers, and in particular about the method of mathematical induction, which will be useful to us in later proofs.

For us, a set will be just a collection of entities, called the elements or members of the set. We indicate that some object x is an element of a set S by writing xES. If x is not an element of S, we write x f£ S.

In order to specify a set S, we must indicate which objects are elements of S. If S is finite, we can do this by writing down all the elements inside braces. For example, we write

S={1,2,3,4}

to signify that S consists of the positive integers 1,2,3, and 4. If S is infinite, then we cannot list all its elements, but sometimes we can give enough of them to make it clear what set S is. For instance,

S= {1,4, 7,10,13,16, ... }

indicates the set of all positive integers that are of the form 1 + 3k for some nonnegative integer k.

We can also specify a set by giving a criterion that determines which objects are in the set. Using this method, the set {l,2,3,4} could be denoted by

{xix is a positive integer ~4},

where the vertical bar stands for the words "such that." Likewise, the set {1,4,7, 10, 13, 16, ... } could be written as

{xix = 1 + 3k for some nonnegative integer k}.

1

2 Section O. Sets and Induction

Some sets occur so frequently that it IS worthwhile to adopt special notations for them. For example, we use

Z to denote the set of all integers,

Q to denote the set of all rational numbers,

R to denote the set of all real numbers, and

C to denote the set of all complex numbers.

The symbol 0 denotes the empty set or null set, i.e., the set with no elements. Sometimes we wish to express the fact that one set is included in another,

i.e., that every element of the first set is also an element of the second set. We do so by saying that the first set is a subset of the second.

DEFINITION If Sand T are sets, then we say that S is a subset of T, and write S k T, if every element of S is an element of T.

Examples If S={1,2,3} and T={l,2,3,4,5}, then SkT. If S= {7T, V2} and T= {7T,5, V2 },