Aimed at those acquainted with basic point-set topology and algebra, this text goes up to the frontiers of current research in topological fields (more precisely, topological rings that algebraically are fields). The reader is given enough background to tackle the current literature without undue additional preparation. Many results not in the text (and many illustrations by example of theorems in the text) are included among the exercises. Sufficient hints for the solution of the exercises are offered so that solving them does not become a major research effort for the reader. A comprehensive bibliography completes the volume.
Author(s): S. Warner
Series: Notas De Matematics, Vol 126
Publisher: Elsevier Science Ltd
Year: 1989
Language: English
Pages: 579
Topological Fields......Page 4
Copyright Page......Page 5
Foreword......Page 8
Notation......Page 10
Table of Contents......Page 12
1. Topological Groups......Page 16
Exercises......Page 23
2. Subgroups......Page 25
Exercises......Page 29
3. Quotient Groups......Page 30
Exercises......Page 37
4. Complete Groups......Page 39
Exercises......Page 46
5. The Bilateral Completion of a Hausdorff Group......Page 47
Exercises......Page 54
6. Metrizable Groups......Page 55
Exercises......Page 64
7. Metric Topologies on Groups......Page 65
Exercises......Page 70
8. Closed Graph and Open Mapping Theorems......Page 71
Exercises......Page 76
9. Locally Compact Topologies on Groups......Page 78
Exercises......Page 90
11. Topological Rings......Page 92
Exercises......Page 98
12. Topological Modules......Page 100
Exercises......Page 107
13. Completions of Topological Rings and Modules......Page 110
Exercises......Page 115
14. Continuity of Inversion......Page 121
Exercises......Page 128
15. Locally Bounded Modules......Page 129
16. Normed and Locally Bounded Rings......Page 133
Exercises......Page 140
17. Normable Rings......Page 142
Exercises......Page 149
18. Values......Page 152
Exercises......Page 161
19. Topologies Definable by Absolute Values......Page 162
Exercises......Page 173
20. Valuations......Page 175
Exercises......Page 191
21. Discrete Valuations......Page 196
Exercises......Page 208
22. An Introduction to Nonarchimedean Analysis......Page 209
Exercises......Page 218
23. Topological Vector Spaces over Valued Division Rings......Page 222
Exercises......Page 234
24. Finite-dimensional Vector Spaces......Page 239
Exercises......Page 246
25. Principles of Functional Analysis......Page 251
Exercises......Page 262
26. Extensions of Absolute Values......Page 271
Exercises......Page 281
27. Locally Compact Division Rings......Page 283
Exercises......Page 286
28. Approximation Theorems......Page 290
Exercises......Page 302
29. Extensions of Valuations......Page 305
Exercises......Page 313
30. Valuations on Algebraic Extensions......Page 314
Exercises......Page 332
31. Maximal Valuations and Linear Compactness......Page 338
Exercises......Page 350
32. Henselian Valuations......Page 360
Exercises......Page 378
33. Locally Bounded Topologies on the Rational Field......Page 396
Exercises......Page 412
34. Dedekind Domains......Page 413
Exercises......Page 431
35. Linear Topologies on the Quotient Field of a Dedekind Domain......Page 432
Exercises......Page 445
36. Locally Bounded Topologies on Algebraic Number Fields and Algebraic Function Fields......Page 446
Exercises......Page 464
37. Locally Bounded Topologies on Orders of Algebraic Number Fields and Algebraic Function Fields......Page 468
Exercises......Page 481
38. The Origin of the Theory of Topological Fields......Page 484
39. Absolute Values......Page 487
40. Valuation Theory......Page 492
41. Topological Vector Spaces......Page 500
42. Topological Groups......Page 505
43. Norms......Page 511
44. Locally Bounded Topologies......Page 517
Bibliography......Page 526
Name Index......Page 564
Subject Index......Page 570
