Notes on Galois Theory

Basic ring theory......Page 3
Some applications of Zorn's lemma......Page 5
Polynomial Rings......Page 7
Polynomials over Q......Page 9
Finite fields......Page 11
Extensions of rings and fields......Page 14
Symmetric polynomials......Page 15
Integral ring extensions......Page 17
Prime ideals in Z[x]: elementary classification......Page 19
The spectrum of a commutative ring......Page 21
`39`42`"613A``45`47`"603ASpec(Z[x])......Page 22
Algebraic field extensions......Page 23
The ring of algebraic integers and the field of algebraic numbers......Page 24
Field extensions of finite degree......Page 25
Some abelian numbers......Page 26
Constructible numbers......Page 27
Splitting fields......Page 29
Automorphisms of finite extensions......Page 33
Galois extensions......Page 34
The Galois correspondence......Page 36
The Galois group of a polynomial......Page 37
Imprimitive group actions and Galois groups......Page 39
The Primitive Element Theorem......Page 40
Galois' view of Galois groups......Page 41
Transitive subgroups......Page 43
Invariant Theory and Resolvents......Page 45
The discriminant......Page 46
Cubic Polynomials......Page 48
Quartic Polynomials......Page 49
The ring of integers in a number field......Page 54
Decomposition and inertia groups......Page 57
Frobenius classes in the Galois group of a polynomial......Page 59
Cyclotomic extensions and abelian numbers......Page 61
Gauss and Cyclotomy......Page 62
The Kronecker-Weber theorem and abelian numbers......Page 66