Galois Theory - Emil Artin [algebra]

D R . E M I L ARTIN
Professor of Mathematics, Princeton University

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
J.
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 . . . . . . . . . . . . . . . . . . . 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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