Fields and Galois theory

Contents......Page 1
References.......Page 3
Fields......Page 4
The characteristic of a field......Page 5
Review of polynomial rings......Page 6
Factoring polynomials......Page 7
Extension fields......Page 10
Construction of some extension fields......Page 11
The subring generated by a subset......Page 12
Algebraic and transcendental elements......Page 13
Transcendental numbers......Page 15
Constructions with straight-edge and compass.......Page 17
Algebraically closed fields......Page 20
Exercises 1--4......Page 21
Maps from simple extensions.......Page 22
Splitting fields......Page 23
Multiple roots......Page 25
Exercises 5--10......Page 27
Groups of automorphisms of fields......Page 29
Separable, normal, and Galois extensions......Page 31
The fundamental theorem of Galois theory......Page 33
Examples......Page 36
Constructible numbers revisited......Page 37
Solvability of equations......Page 38
Exercises 11--13......Page 39
When is GfAn?......Page 40
When is Gf transitive?......Page 41
Quartic polynomials......Page 42
Examples of polynomials with Sp as Galois group over Q......Page 44
Finite fields......Page 45
Computing Galois groups over Q......Page 46
Exercises 14--20......Page 49
Primitive element theorem.......Page 50
Fundamental Theorem of Algebra......Page 52
Cyclotomic extensions......Page 53
Independence of characters......Page 56
The normal basis theorem......Page 57
Hilbert\'s Theorem 90.......Page 58
Cyclic extensions.......Page 60
Proof of Galois\'s solvability theorem......Page 61
The general polynomial of degree n......Page 63
Norms and traces......Page 66
Exercises 21--23......Page 70
Zorn\'s Lemma......Page 71
Third proof of the existence of algebraic closures......Page 72
(Non)uniqueness of algebraic closures......Page 73
Infinite Galois extensions......Page 75
Transcendental extensions......Page 77
Review exercises......Page 82
Solutions to Exercises......Page 87
Two-hour Examination......Page 95
Solutions......Page 96
Index......Page 98