Galois Theory 2nd ed. - E. Artin.pdf

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GALOIS THEORY
353117
NOTRE DAME MATHEMATICAL LECTURES
Number 2
GALOIS THEORY
Lectures delivered at the University of Notre Dame
by
DR. EMIL ARTIN
Professor of Mathematics, Princeton University
Edited and supplemented with a Section on Applications
by
DR. ARTHUR N. MILGRAM
Associate Professor of Mathematics, University of Minnesota
Second Edition
With Additions and Revisions
UNIVERSITY OF NOTRE DAME PRESS
NOTRE DAME
LONDON
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Copyright 1942, 1944
UNIVERSITY OF NOTRE DAME
Second Printing, February 1964
Third Printing, July 1965
Fourth Printing, August 1966
New composition with corrections
Fifth Printing, March 1970
Sixth Printing, January 197 1
Printed in the United States of America by
NAPCO Graphie Arts, Inc., Milwaukee, Wisconsin
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TABLE OF CONTENTS
(The sections marked with an asterisk
have been herein added to the content
of the first edition)
Page
1 LINEAR ALGEBRA .................................... 1
A. Fields........................................... 1
B. Vector Spaces .................................... 1
C. Homogeneous Linear Equations ..................... 2
D. Dependence and Independence of Vectors .. , ......... 4
E. Non-homogeneous Linear Equations ................. 9
F.* Determinants ..................................... 11
II FIELD THEORY ............................. < ......... 21
A. Extension Fields .................................
21
B. Polynomials ......................................
22
C. Algebraic Elements ...............................
25
D. Splitting Fields ...................................
30
E. Unique Decomposition of Polynomials
into Irreducible Factors ........... , .......... 33
F. Group Characters ................................. 34
G.* Applications and Examples to Theorem 13 ............ 38
H. Normal Extensions ................................
41
Finite Fields ............................... . .... 49
Roots of Unity ............................. . . .., .. 56
K. Noether Equations ................................ 57
L. Kummer’s Fields ....................... . .......... 59
M. Simple Extensions ................................ 64
N. Existence of a Normal Basis ........... , ........... 66
Q. Theorem on Natural Irrationalities ...................
J.
67
111 APPLICATIONS
By A. N. Milgram., ..................... , ........... 69
A. Solvable Groups ..................................
69
B. Permutation Groups ...............................
70
C. Solution of Equations by Radicals ...................
72
D. The General Equation of Degree n. ..................
74
E. Solvable Equations of Prime Degree .................
76
F. Ruler and Compass Construction ....................
80
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1 LINEAR ALGEBRA
--*
A field is a set of elements in which a pair of operations called
multiplication and addition is defined analogous to the operations of
multipl:ication and addition in the real number system (which is itself
an example of a field). In each field F there exist unique elements
called o and 1 which, under the operations of addition and multiplica-
tion, behave with respect to a11 the other elements of F exactly as
their correspondents in the real number system. In two respects, the
analogy is not complete: 1) multiplication is not assumed to be commu-
tative in every field, and 2) a field may have only a finite number
of elements.
More exactly, a field is a set of elements which, under the above
mentioned operation of addition, forms an additive abelian group and
for which the elements, exclusive of zero, form a multiplicative group
and, finally, in which the two group operations are connected by the
distributive law. Furthermore, the product of o and any element is de-
fined to be o.
If multiplication in the field is commutative, then the field is
called a commutative field.
B. Vector Spaces.
If V is an additive abelian group with elements A, B, . . . ,
F a field with elements a, b, . . . , and if for each a c F and A e V
A. Fie’lds
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