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Algebra : Volume II: Fields with Structure, Algebras and Advanced Topics - Falko Lorenz


Volume II: Fields with Structure, Algebras and Advanced Topics

By: Falko Lorenz, Silvio Levy (Translator)


Published: 27th December 2007
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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.

From the reviews: "Algebra II: Fields with Structure, Algebras and Advanced Topics is ! a complete algebra course, including both undergraduate and graduate topics. ! The second volume focuses on fields with structure and algebras. ! the choice of topics and their organization are excellent and provide a unifying view of most of algebra. In all, Lorenz's book is a wonderful reference for both teachers and researches, and can be used with much profit for independent study by hard-working students." (Luiz Hendrique de Figueiredo, MathDL, July, 2008) "The author has managed to cover such an amazing wealth of advanced material in a very adroit manner, thereby keeping the representation utmost lively, comprehensible, thorough, always straight to the point, essentially self-contained, methodologically elegant and -- all in all -- admirably reader-friendly. ! No doubt, this is an outstanding textbook on advanced topics in abstract algebra, mainly in its field-theoretic and number-theoretic aspects, and its availability in English makes it a unique and utmost valuable enrichment of the international textbook literature on the subject." (Werner Kleinert, Zentralblatt MATH, Vol. 1130 (8), 2008)

Forewordp. v
Ordered Fields and Real Fieldsp. 1
Ordered and preordered fieldsp. 2
Extensions of field ordersp. 4
Real-closed fieldsp. 5
The fundamental theorem of algebrap. 7
Artin's characterization of real-closed fieldsp. 8
Sylvester's theorem on the number of real rootsp. 10
Extension of order-preserving homomorphismsp. 11
Existence of real specializationsp. 12
Hilbert's Seventeenth Problem and the Real Nullstellensatzp. 15
Artin's solution to Hilbert's seventeenth problemp. 15
Generalization to affine K-varietiesp. 16
The real Nullstellensatzp. 18
Positive definite functions on semialgebraic setsp. 23
Positive definite symmetric functionsp. 25
Orders and Quadratic Formsp. 29
Witt equivalence; the Witt ring W (K) and its prime idealsp. 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 Fieldsp. 39
Absolute values on the rationalsp. 40
Nonarchimedean absolute valuesp. 45
Completion of absolute valuesp. 47
The field Q[subscript p] of p-adic numbersp. 52
Equivalence of normsp. 54
Hensel's Lemmap. 56
Extension of absolute valuesp. 59
Residue Class Degree and Ramification Indexp. 65
Discrete absolute valuesp. 67
The formula ef = n for complete, discrete valuationsp. 68
Unramified extensionsp. 70
Purely ramified extensionsp. 73
Local Fieldsp. 75
Classification of local fieldsp. 76
Connection with global fieldsp. 78
Completion of the algebraic closure of Q[subscript p]p. 80
Solvability of Galois groupsp. 81
Structure of the multiplicative groupp. 83
The case of Q[subscript p]p. 91
Witt Vectorsp. 93
Teichmuller representativesp. 94
Ghost componentsp. 96
Witt's Lemmap. 98
The ring of Witt vectorsp. 100
Higher Artin-Schreier theoryp. 105
The Tsen Rank of a Fieldp. 109
Tsen's theoremsp. 110
Behavior of the Tsen rank with respect to extensionsp. 112
Norm formsp. 114
C[subscript i]-fieldsp. 116
The Lang-Nagata Theoremp. 118
Finite fields have Tsen rank 1p. 121
The Chevalley-Warning Theoremp. 122
Algebraically closed fields have Tsen rank 0p. 123
Krull's dimension theoremp. 124
Fundamentals of Modulesp. 127
Fundamentals of linear algebrap. 128
Simple and semisimple modulesp. 133
Noetherian and artinian modulesp. 139
The Jordan-Holder Theoremp. 142
The Krull-Remak-Schmidt Theoremp. 143
The Jacobson radicalp. 144
Nilpotence of the radical in artinian ringsp. 148
Artinian algebras are noetherianp. 149
Wedderburn Theoryp. 151
Simple algebrasp. 151
Decomposition of simple algebrasp. 152
Wedderburn's structure theoremp. 155
Tensor products of simple algebrasp. 160
The Brauer group of a fieldp. 164
Tensor products of semisimple algebrasp. 166
The Centralizer Theorem and splitting fieldsp. 168
The Skolem-Noether Theoremp. 173
Reduced norm and tracep. 176
Crossed Productsp. 183
The relative Brauer group of Galois extensionsp. 184
Inflation and restrictionp. 190
The Brauer group is a torsion groupp. 194
Cyclic algebrasp. 196
Quaternion algebrasp. 203
Cohomology groups and the connecting homomorphismp. 212
Corestrictionp. 217
The Brauer Group of a Local Fieldp. 223
Existence of unramified splitting fieldsp. 224
The equality e = f = n for local division algebrasp. 226
The relative Brauer group in the unramified casep. 229
The Hasse invariantp. 231
Consequencesp. 234
Local Class Field Theoryp. 239
The local norm residue symbolp. 240
Functorial properties of the norm residue symbolp. 242
The local reciprocity lawp. 243
The group of universal norms is trivialp. 247
The local existence theoremp. 249
The local Kronecker-Weber theoremp. 251
Semisimple Representations of Finite Groupsp. 253
Terminologyp. 254
Maschke's Theoremp. 259
Applications of Wedderburn theoryp. 261
Orthogonality relations for charactersp. 265
Integrality properties of charactersp. 268
Induced representations and induced charactersp. 272
Artin's induction theoremp. 275
The Brauer-Witt induction theoremp. 277
The Schur Group of a Fieldp. 283
Schur index of absolutely irreducible charactersp. 284
Schur algebrasp. 286
Reduction to cyclotomic algebrasp. 288
The Schur group of a local fieldp. 294
Problems and Remarksp. 299
Indexp. 331
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780387724874
ISBN-10: 0387724877
Series: Universitext : Book 2
Audience: Tertiary; University or College
Format: Paperback
Language: English
Number Of Pages: 340
Published: 27th December 2007
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.63
Weight (kg): 0.53