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 .
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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