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

Emil Artin Notes by Albert A. Blank

15 Algebr a with Galois Theory

Courant Institute of Mathematical Science s New York University New York, New York

American Mathematical Societ y Providence, Rhode Island

http://dx.doi.org/10.1090/cln/015

2000 Mathematics Subject Classification. P r i m a r y 12-01 , 12F10 .

Library o f Congres s Cataloging-in-Publieatio n D a t a

Artin, Emil , 1898-1962 . Algebra wit h Galoi s theor y / E . Artin , note s b y Alber t A . Blank .

p. cm . — (Couran t lectur e note s ; 15 ) ISBN 978-0-8218-4129- 7 (alk . paper ) 1. Galoi s theory . 2 . Algebra . I . Blank , Alber t A . I L Title .

QA214.A76 200 7 512—dc22 200706079 9

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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 Degre e 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 Coefficient s 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

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Titles i n Thi s Serie s

16 S . R . S . Varadhan , Stochasti c processes , 200 7

15 Emi l Art in , Algebr a wit h Galoi s theory , 200 7

14 Pete r D . Lax , Hyperboli c partia l differentia l equations , 200 6

13 Olive r Bühler , A brie f introductio n t o classical , Statistical , an d quantu m mechanics , 200 6

12 Jürge n Mose r an d Eduar d J . Zehnder , Note s o n dynamica l Systems , 200 5

11 V . S . Varadarajan , Supersymmetr y fo r mathematicians : A n introduction , 200 4

10 Thierr y Cazenave , Semilinea r Schrödinge r equations , 200 3

9 Andre w Majda , Introductio n t o PDE s an d wave s fo r th e atmospher e an d ocean , 200 3

8 Fedo r Bogomolo v an d Tihomi r Petrov , Algebrai c curve s an d one-dimensiona l fields ,

2003

7 S . R . S . Varadhan , Probabilit y theory , 200 1

6 Loui s Nirenberg , Topic s i n nonlinea r functiona l analysis , 200 1

5 Emmanue l Hebey , Nonlinea r analysi s o n manifolds : Sobole v Space s an d inequalities ,

2000

3 Perc y Deift , Orthogona l polynomial s an d rando m matrices : A Riemann-Huber t

approach, 200 0

2 Jala l Shata h an d Michae l Struwe , Geometri e wav e equations , 200 0

1 Qin g Ha n an d Fanghu a Lin , Ellipti c partia l differentia l equations , 200 0