Algebra with Galois Theory

137 Pages·2007·24.39 MB·English

by  Emil Artin

Algebra with Galois Theory
Courant Lecture Notes
in Mathematics
Executive Editor
Jalal Shatah
Managing Editor
Paul D. Monsour
Assistant Editor
Reeva Goldsmith
Copy Editor
Marc Nirenberg
http://dx.doi.org/10.1090/cln/015
Emil Artin
Notes by Albert A. Blank
15 Algebr a with Galois Theory
Courant Institute of Mathematical Sciences
New York University
New York, New York
American Mathematical Society
Providence, Rhode Island
2000 Mathematics Subject Classification. Primar y 12-01 , 12F10 .
Library of Congress Cataloging-in-Publieatio n Dat a
Artin, Emil, 1898-1962.
Algebra with Galois theory / E. Artin, notes by Albert A. Blank.
p. cm. — (Courant lecture notes ; 15)
ISBN 978-0-8218-4129-7 (alk. paper)
1. Galois theory. 2 . Algebra. I . Blank, Albert A. I L Title.
QA214.A76 200 7
512—dc22 200706079 9
Printed in the United States of America.
© Th e paper used in this book is acid-free and falls within the guidelines
established to ensure permanence and durability.
Visit the AMS home page at http://www.ams.org /
10 9 8 76 5 4 3 2 1 2 11 10
Contents
Editors' Note
Chapter 1. Group s
1.1. Th e Concept of a Group
1.2. Subgroup s
Chapter 2. Ring s and Fields
2.1. Linea r Equations in a Field
2.2. Vecto r Spaces
Chapter 3. Polynomials . Factorization into Primes. Ideals.
3.1. Polynomial s over a Field
3.2. Factorizatio n into Primes
3.3. Ideal s
3.4. Greates t Common Divisor
Chapter 4. Solutio n of the General Equation of nth Degree
Extension Fields. Isomorphisms.
4.1. Congruenc e
4.2. Extensio n Fields
4.3. Isomorphis m
Chapter 5. Galoi s Theory
5.1. Splittin g Fields
5.2. Automorphism s of the Splitting Field
5.3. Th e Characteristic of a Field
5.4. Derivativ e of a Polynomial: Multiple Roots
5.5. Th e Degree of an Extension Field
5.6. Grou p Characters
5.7. Automorphi c Groups of a Field
5.8. Fundamenta l Theorem of Galois Theory
5.9. Finit e Fields
Chapter 6. Polynomial s with Integral Coefficients
6.1. Irreducibilit y
6.2. Primitiv e Roots of Unity
Chapter 7. Th e Theory of Equations
7.1. Rule r and Compass Constructions
VI CONTENTS
7.2. Solutio n of Equations by Radicals 9 4
7.3. Steinitz ' Theorem 10 4
7.4. Tower s ofFields 10 7
7.5. Permutatio n Groups 11 2
7.6. Abel' s Theorem 12 1
7.7. Polynomial s of Prime Degree 12 3
Editors' Note
Beeause what was in 1947 "modern" has now become Standard, and what was
then "higher" has now become foundational, we have retitled this volume Algebra
with Galois Theory from the original Modern Higher Algebra. Galois Theory.
Jalal Shatah, Executive Editor
Paul Monsour, Managing Editor
August 2007
This page intentionally left blank
http://dx.doi.org/10.1090/cln/015/01
CHAPTER 1
Groups
We concern ourselves with sets G of objects a,b,c,... calle d elements. The
sentence "a is an element of G" will be denoted symbolically by a e G. Assume
an Operation called "multiplication" which assigns to an ordered pair of objects
a, b of G another object a • b (or simply ab) the product of a and b. It is useful to
require that G be closed with respect to multiplication, namely:
(1) lfa.be G,thena-Z> e G.
EXAMPLES.
(a) Let G be the set of positive integers. If subtraction is taken as the "multi-
plication" in G, then G is certainly not closed, e.g., 3-5 = 3 — 5 = —2.
If taking the greatest common divisor is our multiplication, then closure
is obvious.
(b) Take G to be the set of functions of one variable. I f f(x), g(x) e G
dehne f(x) • g(x) = f[g(x)], e.g. , ex • logx = elogx = x.
EXERCISE 1. Write out the multiplication table and thereby show closure for
the set of functions
1
f\=x, fi /3 = 1 u = fs = h
x' 1 — JC' X ~ 1
SOLUTION.
h fr fr fr fr
/l
fr h fr fr fr fr
/l
fr h fr fr fr fr
/i
h h h fr fr fr fr
h U fr fr fr fr fr
h fr fr fr fr fr fr
fr fr fr fr
76 jl
where fi • fj is listed in the /th row and jth column .
We make the further requirement that multiplication obey the associative law:
(2) If a, b,c e G, then (ab)c = a(bc). This is a rather strong condition. It is
not generally satisfied; consider, e.g., subtraction among the integers. For
functions of one variable, as above, it is valid, however. I f f(x), g(x),
h(x) are any three functions we have
(fg)h = f(g(h(x))) = f(gh).
l