Algebra

by Lorenz, Falko; Levy, Silvio (COL)

• ISBN13: 9780387724874
• ISBN10: 0387724877
• Edition: 1st
• Format: Paperback
• Copyright: 2007-12-07
• Publisher: Springer Nature

Summary

The present textbook is a lively, problem-oriented and carefully written introduction to classical modern algebra. The author leads the reader through interesting subject matter, while assuming only the background provided by a first course in linear algebra.

Table of Contents

Foreword	p. v
Ordered Fields and Real Fields	p. 1
Ordered and preordered fields	p. 2
Extensions of field orders	p. 4
Real-closed fields	p. 5
The fundamental theorem of algebra	p. 7
Artin's characterization of real-closed fields	p. 8
Sylvester's theorem on the number of real roots	p. 10
Extension of order-preserving homomorphisms	p. 11
Existence of real specializations	p. 12
Hilbert's Seventeenth Problem and the Real Nullstellensatz	p. 15
Artin's solution to Hilbert's seventeenth problem	p. 15
Generalization to affine K-varieties	p. 16
The real Nullstellensatz	p. 18
Positive definite functions on semialgebraic sets	p. 23
Positive definite symmetric functions	p. 25
Orders and Quadratic Forms	p. 29
Witt equivalence; the Witt ring W (K) and its prime ideals	p. 30
The spectrum of W (K)	p. 32
The torsion elements of W (K)	p. 34
The zero divisors of W (K)	p. 37
Absolute Values on Fields	p. 39
Absolute values on the rationals	p. 40
Nonarchimedean absolute values	p. 45
Completion of absolute values	p. 47
The field Q[subscript p] of p-adic numbers	p. 52
Equivalence of norms	p. 54
Hensel's Lemma	p. 56
Extension of absolute values	p. 59
Residue Class Degree and Ramification Index	p. 65
Discrete absolute values	p. 67
The formula ef = n for complete, discrete valuations	p. 68
Unramified extensions	p. 70
Purely ramified extensions	p. 73
Local Fields	p. 75
Classification of local fields	p. 76
Connection with global fields	p. 78
Completion of the algebraic closure of Q[subscript p]	p. 80
Solvability of Galois groups	p. 81
Structure of the multiplicative group	p. 83
The case of Q[subscript p]	p. 91
Witt Vectors	p. 93
Teichmuller representatives	p. 94
Ghost components	p. 96
Witt's Lemma	p. 98
The ring of Witt vectors	p. 100
Higher Artin-Schreier theory	p. 105
The Tsen Rank of a Field	p. 109
Tsen's theorems	p. 110
Behavior of the Tsen rank with respect to extensions	p. 112
Norm forms	p. 114
C[subscript i]-fields	p. 116
The Lang-Nagata Theorem	p. 118
Finite fields have Tsen rank 1	p. 121
The Chevalley-Warning Theorem	p. 122
Algebraically closed fields have Tsen rank 0	p. 123
Krull's dimension theorem	p. 124
Fundamentals of Modules	p. 127
Fundamentals of linear algebra	p. 128
Simple and semisimple modules	p. 133
Noetherian and artinian modules	p. 139
The Jordan-Holder Theorem	p. 142
The Krull-Remak-Schmidt Theorem	p. 143
The Jacobson radical	p. 144
Nilpotence of the radical in artinian rings	p. 148
Artinian algebras are noetherian	p. 149
Wedderburn Theory	p. 151
Simple algebras	p. 151
Decomposition of simple algebras	p. 152
Wedderburn's structure theorem	p. 155
Tensor products of simple algebras	p. 160
The Brauer group of a field	p. 164
Tensor products of semisimple algebras	p. 166
The Centralizer Theorem and splitting fields	p. 168
The Skolem-Noether Theorem	p. 173
Reduced norm and trace	p. 176
Crossed Products	p. 183
The relative Brauer group of Galois extensions	p. 184
Inflation and restriction	p. 190
The Brauer group is a torsion group	p. 194
Cyclic algebras	p. 196
Quaternion algebras	p. 203
Cohomology groups and the connecting homomorphism	p. 212
Corestriction	p. 217
The Brauer Group of a Local Field	p. 223
Existence of unramified splitting fields	p. 224
The equality e = f = n for local division algebras	p. 226
The relative Brauer group in the unramified case	p. 229
The Hasse invariant	p. 231
Consequences	p. 234
Local Class Field Theory	p. 239
The local norm residue symbol	p. 240
Functorial properties of the norm residue symbol	p. 242
The local reciprocity law	p. 243
The group of universal norms is trivial	p. 247
The local existence theorem	p. 249
The local Kronecker-Weber theorem	p. 251
Semisimple Representations of Finite Groups	p. 253
Terminology	p. 254
Maschke's Theorem	p. 259
Applications of Wedderburn theory	p. 261
Orthogonality relations for characters	p. 265
Integrality properties of characters	p. 268
Induced representations and induced characters	p. 272
Artin's induction theorem	p. 275
The Brauer-Witt induction theorem	p. 277
The Schur Group of a Field	p. 283
Schur index of absolutely irreducible characters	p. 284
Schur algebras	p. 286
Reduction to cyclotomic algebras	p. 288
The Schur group of a local field	p. 294
Problems and Remarks	p. 299
Index	p. 331
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