Skew fields: theory of general division rings

Cover......Page 1
ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS......Page 2
Title......Page 4
Copyright......Page 5
Dedication......Page 6
CONTENTS......Page 8
PREFACE......Page 12
From the preface to Skew Field Constructions......Page 14
NOTE TO THE READER......Page 16
PROLOGUE......Page 18
1.1 Fields, skew fields and near fields......Page 20
1.2 The general embedding problem......Page 25
1.3 Ore\'s method......Page 31
1.4 Necessary conditions for a field of fractions to exist......Page 36
1.5 Stable association and similarity......Page 41
1.6 Free algebras, firs and semifirs......Page 51
1.7 The matrix reduction functor......Page 58
Notes and comments......Page 62
2.1 Skew polynomial rings......Page 64
2.2 The ideal structure of skew polynomial rings......Page 73
2.3 Power series rings......Page 83
2.4 Group rings and the Malcev-Neumann construction......Page 88
2.5 Iterated skew polynomial rings......Page 95
2.6 Fields of fractions for a class of filtered rings......Page 100
Notes and comments......Page 108
3.1 The degree of a field extension......Page 110
3.2 The Jacobson-Bourbaki correspondence......Page 113
3.3 Galois theory......Page 117
3.4 Equations over skew fields and Wedderburn\'s theorem......Page 128
3.5 Pseudo-linear extensions......Page 135
3.6 Quadratic extensions......Page 143
3.7 Outer cyclic Galois extensions......Page 150
3.8 Infinite outer Galois extensions......Page 157
3.9 The multiplicative group of a skew field......Page 160
Notes and comments......Page 167
4 Localization......Page 169
4.1 The category of epic R-fields and specializations......Page 170
4.2 The matrix representation of fractions......Page 174
4.3 The construction of the localization......Page 179
4.4 Matrix ideals......Page 188
4.5 Universal fields of fractions......Page 194
4.6 Projective rank functions and hereditary rings......Page 201
4.7 Normal forms for matrix blocks over firs......Page 213
Notes and comments......Page 217
5 Coproducts of fields......Page 219
5.1 The coproduct construction for groups and rings......Page 220
5.2 Modules over coproducts......Page 227
5.3 Submodules of induced modules over a coproduct......Page 231
5.4 The tensor ring on a bimodule......Page 241
5.5 HNN-extension of fields......Page 248
5.6 HNN-extensions of rings......Page 256
5.7 Adjoining generators and relations......Page 259
5.8 Derivations......Page 267
5.9 Field extensions with different left and right degrees......Page 276
5.10 Coproducts of quadratic field extensions......Page 285
Notes and comments......Page 292
6 General skew fields......Page 295
6.1 Presentations of skew fields......Page 296
6.2 The specialization lemma......Page 300
6.3 Normal forms for matrices over a tensor ring......Page 309
6.4 Free fields......Page 318
6.5 Existentially closed skew fields......Page 325
6,6 The word problem for skew fields......Page 334
6.7 The class of rings embeddable in fields......Page 343
Notes and comments......Page 346
7.1 Polynomial identities......Page 348
7.2 Rational identities......Page 352
7.3 Specializations......Page 357
7.4 A particular rational identity for matrices......Page 362
7.5 The rational meet of a family of Z-rings......Page 365
7.6 The support relation on generic division algebras......Page 372
7.7 Examples of support relations......Page 379
Notes and Comments......Page 382
8 Equations and singularities......Page 383
8.1 Algebraically closed skew fields......Page 384
8.2 Left and right eigenvalues of a matrix......Page 391
8.3 Normal forms for a single matrix over a skew field......Page 396
8.4 Central localizations of polynomial rings......Page 403
8.5 The solution of equations over skew fields......Page 412
8.6 Specializations and the rational topology......Page 418
8.7 Examples of singularities......Page 425
8.8 Nullstellensatz and elimination......Page 428
Notes and comments......Page 435
9 Valuations and orderings on skew fields......Page 437
9.1 The basic definitions......Page 438
9.2 Abelian and quasi-commutative valuations......Page 444
9.3 Matrix valuations on rings......Page 452
9 A Subvaluations and matrix subvaluations......Page 459
9.5 Matrix valuations on firs......Page 466
9.6 Ordered rings and fields......Page 474
9.7 Matrix cones and orderings on skew fields......Page 479
Notes and comments......Page 486
STANDARD NOTATIONS......Page 490
LIST OF SPECIAL NOTATIONS USED THROUGHOUT THE TEXT......Page 492
BIBLIOGRAPHY AND AUTHOR INDEX......Page 495
SUBJECT INDEX......Page 512