Classics in Mathematics

By: H.G. Zimmer (Translator), Helmut Hasse

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Paperback | 10 January 2002

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664 Pages

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"...a fine book ... treats algebraic number theory from the valuation-theoretic viewpoint. When it appeared in 1949 it was a pioneer. Now there are plenty of competing accounts. But Hasse has something extra to offer. This is not surprising, for it was he who inaugurated the local-global principle (universally called the Hasse principle). This doctrine asserts that one should first study a problem in algebraic number theory locally, that is, at the completion of a vaulation. Then ask for a miracle: that global validity is equivalent to local validity. Hasse proved that miracles do happen in his five beautiful papers on quadratic forms of 1923-1924. ... The exposition is discursive. ... It is trite but true: Every number-theorist should have this book on his or her shelf."
(Irving Kaplansky in Bulletin of the American Mathematical Society, 1981)

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Non-Fiction Mathematics Number Theory

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Non-Fiction Mathematics Number Theory

The Foundations of Arithmetic in the Rational Number Field	p. 1
Prime Decomposition	p. 2
Divisibility	p. 8
Congruences	p. 24
The Structure of the Residue Class Ring mod m and of the Reduced Residue Class Group mod m	p. 42
Quadratic Residues	p. 64
The Theory of Valued Fields	p. 105
The Fundamental Concepts Regarding Valuations	p. 105
Arithmetic in a Discrete Valued Field	p. 129
The Completion of a Valued Field	p. 136
The Completion of a Discrete Valued Field. The p-adic Number Fields	p. 144
The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class Field	p. 151
Prolongation of a Discrete Valuation to a Purely Transcendental Extension	p. 176
Prolongation of the Valuation of a Complete Field to a Finite-Algebraic Extension	p. 182
The Isomorphism Types of Complete Archimedean Valued Fields	p. 191
The Structure of a Finite-Algebraic Extension of a Complete Discrete Valued Field	p. 194
The Structure of the Multiplicative Group of a Complete Discrete Valued Field with Perfect Residue Class Field of Prime Characteristic	p. 213
The Tamely Ramified Extension Types of a Complete Discrete Valued Field with Finite Residue Class Field of Characteristic p	p. 248
The Exponential Function, the Logarithm, and Powers in a Complete Non-Archimedean Valued Field of Characteristic O	p. 255
Prolongation of the Valuation of a Non-Complete Field to a Finite-Algebraic Extension	p. 276
The Foundations of Arithmetic in Algebraic Number Fields	p. 309
Relations Between the Complete System of Valuations and the Arithmetic of the Rational Number Field	p. 310
Prolongation of the Complete System of Valuations to a Finite-Algebraic Extension	p. 324
The Prime Spots of an Algebraic Number Field and their Completions	p. 335
Decomposition into Prime Divisors, Integrality, and Divisibility	p. 338
Congruences	p. 367
The Multiples of a Divisor	p. 372
Differents and Discriminants	p. 429
Quadratic Number Fields	p. 494
Cyclotomic Fields	p. 520
Units	p. 536
The Class Number	p. 562
Approximation Theorems and Estimates of the Discriminant	p. 596
Index of Names	p. 628
Subject Index	p. 629
Table of Contents provided by Blackwell. All Rights Reserved.

Classics in Mathematics

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