文档介绍：ABSTRACT ALGEBRA:
REVIEW PROBLEMS
ON GROUPS AND
GALOIS THEORY
John A. Beachy
Northern Illinois University
2000
ii
This is a supplement to
Abstract Algebra, Second Edition
by John A. Beachy and William D. Blair
ISBN 0–88133–866–4, Copyright 1996
Waveland Press, Inc.
. Box 400
Prospect Heights, Illinois 60070
847 / 634-0081
c John A. Beachy 2000
Permission is granted to copy this document in electronic form, or to print it for
personal use, under these conditions:
it must be reproduced in whole;
it must not be modified in any way;
it must not be used as part of another publication.
Formatted October 15, 2002, at which time the original was available at:
/∼beachy/abstract algebra/
Contents
PREFACE iv
7 STRUCTURE OF GROUPS 1
Some Examples 2
Isomorphism theorems 8
Conjugacy 12
Group actions 13
The Sylow theorems 16
Finite abelian groups 17
Solvable groups 19
Simple groups 20
8 GALOIS THEORY 23
Splitting fields 23
Galois groups 27
Repeated roots 29
The fundamental theorem 30
Solvability by radicals 32
SOLVED PROBLEMS: 34
7 Group Theory Solutions 35
8 Galois Theory Solutions 51
BIBLIOGRAPHY 61
INDEX 62
iii
iv PREFACE
PREFACE
My goal is to provide some help in reviewing Chapters 7 and 8 of our book
Abstract Algebra. I have included summaries of most of these sections, together
with some ments. The review problems are intended to have relatively
short answers, and to be more typical of exam questions than of standard textbook
exercises.
By assuming that this is a review, I have been able make some minor changes
in the order of presentation. The first section covers various examples of groups.
In presenting these examples, I have introduced some concepts that are not studied
until later in the text. I think it is helpful to have the examples collected in one
spot, so that you can refer to them as you review.
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