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Field Theory: Classical Foundations and Multiplicative Groups

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Provides summary of field theory that emphasizes refinements and extensions achieved in recent studies. It describes canonical fundamental units of certain classes of pure cubic fields, proves Knesser's theorem on torsion groups of separable field extensions, establishes a theorem that provides nece

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Provides summary of field theory that emphasizes refinements and extensions achieved in recent studies. It describes canonical fundamental units of certain classes of pure cubic fields, proves Knesser's theorem on torsion groups of separable field extensions, establishes a theorem that provides necessary and sufficient conditions for the Galois group of a binomial Xn-a to be abelian, shows the result of the degrees of sums in a separable field extension. Annotation c. Book News, Inc., Portland, OR (
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Product Details

Table of Contents

Preface v
Chapter 1. Preliminaries 1
1. Notation and terminology 1
2. Polynomial algebras 7
3. Integral extensions 29
4. Tensor products 32
5. Module-theoretic prerequisites 37
6. Topological prerequisites 41
Chapter 2. Classical Topics in Field Theory 47
1. Algebraic extensions 47
2. Normal extensions 57
3. Separable, purely inseparable and simple extensions 63
4. Galois extensions 79
5. Finite fields, roots of unity and cyclotomic extensions 85
6. Norms, traces and their applications 103
7. Discriminants and integral bases 117
8. Units in quadratic fields 135
9. Units in pure cubic fields 148
10. Finite Galois theory 167
11. Profinite groups 172
12. Infinite Galois theory 184
13. Witt vectors 192
14. Cyclic extensions 205
15. Kummer theory 214
16. Radical extensions and related results 221
17. Degrees of sums in a separable field extension 242
18. Galois cohomology 247
19. The Brauer group of a field 263
20. An interpretation of H[superscript 3 subscript O](G,E*) 282
21. A cogalois theory for radical extensions 303
22. Abelian p-extensions over fields of characteristic p 323
23. Formally real fields 333
24. Transcendental extensions 348
Chapter 3. Valuation Theory 353
1. Valuations 353
2. Valuation rings and places 368
3. Dedekind domains 376
4. Completion of a field 389
5. Extensions of valuations 400
6. Valuations of algebraic number fields 411
7. Ramification index and residue degree 414
8. Structure of complete discrete valued fields 421
A. Notation and terminology 421
B. The equal characteristic case 422
C. The unequal characteristic case 427
D. The inertia field 431
E. Cyclotomic extensions of p-adic fields 436
Chapter 4. Multiplicative Groups of Fields 439
1. Some general observations 439
2. Infinite abelian groups 443
3. The Dirichlet-Chevalley-Hasse Unit Theorem 449
4. The torsion subgroup 461
5. Global fields 463
6. Algebraically closed, real closed and the rational p-adic fields 468
7. Local fields 474
A. Preparatory results 475
B. The equal characteristic case 481
C. The unequal characteristic case 482
8. Extensions of algebraic number fields 487
9. Brandis's theorem 496
10. Fields with free multiplicative groups modulo torsion 501
11. A nonsplitting example 517
12. Embedding groups 519
13. Multiplicative groups under field extensions 525
14. Notes 531
Bibliography 535
Notation 541
Index 547
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