Algebra. Fields with structure, algebras and advanced topics

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Edition: 1

Series: Universitext

Volume: Volume 2

ISBN: 0387724877, 9780387724874

Size: 2 MB (2318242 bytes)

Pages: 343/343

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Falko Lorenz, Silvio Levy0387724877, 9780387724874

From Math Reviews: This is Volume II of a two-volume introductory text in classical algebra. The text moves carefully with many details so that readers with some basic knowledge of algebra can read it without difficulty. The book can be recommended either as a textbook for some particular algebraic topic or as a reference book for consultations in a selected fundamental branch of algebra. The book contains a wealth of material. Amongst the topics covered in Volume II the reader can find: the theory of ordered fields (e.g., with reformulation of the fundamental theorem of algebra in terms of ordered fields, with Sylvester’s theorem on the number of real roots), Nullstellen-theorems (e.g., with Artin’s solution of Hilbert’s 17th problem and Dubois’ theorem), fundamentals of the theory of quadratic forms, of valuations, local fields and modules. The book also contains some lesser known or nontraditional results; for instance, Tsen’s results on solubility of systems of polynomial equations with a sufficiently large number of indeterminates. These two volumes constitute a very good, readable and comprehensive survey of classical algebra and present a valuable contribution to the literature on this subject.

Table of contents :
Foreword……Page 5
Contents……Page 7
Ordered Fields and Real Fields……Page 10
Hilbert’s Seventeenth Problem and the Real Nullstellensatz……Page 24
Orders and Quadratic Forms……Page 37
Absolute Values on Fields……Page 47
Residue Class Degree and Ramification Index……Page 72
Local Fields……Page 82
Witt Vectors……Page 99
The Tsen Rank of a Field……Page 115
Fundamentals of Modules……Page 133
Wedderburn Theory……Page 156
Crossed Products……Page 187
The Brauer Group of a Local Field……Page 227
Local Class Field Theory……Page 243
Semisimple Representations of Finite Groups……Page 256
The Schur Group of a Field……Page 285
Appendix:Problems and Remarks……Page 302
Index……Page 334

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