The Structure of Fields

The Structure of Fields [David Winter]......Page Cover_0001.djvu
Title......Page 001_0001.djvu
Copyright......Page 002_0001.djvu
Dedication......Page 003_0001.djvu
Preface......Page 005_0001.djvu
Table of Contents......Page 009_0001.djvu
0.1 Basic algebra......Page 013_0001.djvu
0.2 Groups......Page 018_0001.djvu
0.3 Transformation groups......Page 021_0001.djvu
0.4 The Krull closure in a group G......Page 025_0001.djvu
Exercises......Page 026_0001.djvu
1.1 Preliminaries......Page 038_0001.djvu
1.2 Algebraic extensions......Page 041_0001.djvu
1.3 Splitting fields......Page 043_0001.djvu
1.4 Algebraic closure......Page 046_0001.djvu
1.5 Finite fields......Page 048_0001.djvu
1.6 Transcendency basis of a field extension......Page 050_0001.djvu
Exercises......Page 053_0001.djvu
2.1 The structure of an irreducible polynomial......Page 061_0001.djvu
2.2 Separable and radical extensions......Page 062_0001.djvu
2.3 Normal and Galois extensions......Page 068_0001.djvu
2.4 Composites......Page 070_0001.djvu
Exercises......Page 073_0001.djvu
3.1 Linear independence of homomorphisms......Page 077_0001.djvu
3.2 Galois descent......Page 078_0001.djvu
3.3 The Galois correspondence theorem......Page 081_0001.djvu
3.4 The normal basis theorem......Page 085_0001.djvu
3.5 Algebraic independence of homomorphisms......Page 086_0001.djvu
3.6 Norm and trace......Page 087_0001.djvu
3.7 Galois cohomology......Page 088_0001.djvu
3.8 Cyclotomic extensions......Page 091_0001.djvu
3.9 Cyclic extensions......Page 093_0001.djvu
3.10 Abelian extensions......Page 098_0001.djvu
3.11 Solvable extensions......Page 100_0001.djvu
3.12 Theory of equations......Page 102_0001.djvu
Exercises......Page 103_0001.djvu
4.1 Algebraic function fields and their geometrical interpretation......Page 109_0001.djvu
4.2 Algebraic function fields of transcendency degree 1......Page 111_0001.djvu
4.3 Separably generated algebraic function fields......Page 112_0001.djvu
Exercises......Page 119_0001.djvu
5.1 Rings of endomorphisms of K......Page 123_0001.djvu
5.2 Lie rings of derivations of K......Page 126_0001.djvu
5.3 Birings of endomorphisms of K......Page 130_0001.djvu
Exercises......Page 135_0001.djvu
6.1 Tori......Page 138_0001.djvu
6.2 Diagonalizable toral k-subrings of H(K/k)......Page 140_0001.djvu
6.3 Coradical toral k-subrings of H(K/k)......Page 143_0001.djvu
6.4 Radical extensions......Page 149_0001.djvu
Exercises......Page 152_0001.djvu
S.1 Sets and elements......Page 162_0001.djvu
S.2 Functions......Page 163_0001.djvu
S.3 Cardinality......Page 164_0001.djvu
S.5 transfinite induction......Page 165_0001.djvu
T.1 Preliminaries......Page 166_0001.djvu
T.2 Tensor products of linear transformations......Page 169_0001.djvu
T.3 The identity, commutative, associative and distributive laws for tensor products......Page 170_0001.djvu
T.4 Ascent and descent......Page 172_0001.djvu
W. Witt vectors......Page 174_0001.djvu
A.1 Preliminaries......Page 180_0001.djvu
A.2 Tensor products of algebras......Page 181_0001.djvu
A.3 Finite dimensional commutative algebras......Page 182_0001.djvu
C.1 Preliminaries......Page 185_0001.djvu
C.3 Duality......Page 187_0001.djvu
C.4 Cocommutative coalgebras......Page 191_0001.djvu
B.1 Preliminaries......Page 195_0001.djvu
B.2 Conormal bialgebras......Page 198_0001.djvu
B.3 Tensor products and semidirect products......Page 199_0001.djvu
B.4 Measuring bialgebras......Page 200_0001.djvu
B.5 Bialgebras and the structure of finite dimensional field extensions......Page 204_0001.djvu
References......Page 209_0001.djvu
Index of Symbols......Page 211_0001.djvu
Index of Terminology......Page 213_0001.djvu