Number Theory

Helmut Hasse

638 pages, parution le 05/02/2002

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Résumé

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 theoy 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.

Contents

The Foundations of Arithmetic in the Ratioanl Number Field

Prime Decomposition
Divisibility
Congruences
The Structure of the Residue Class Ring mod m and of the Reduced Residue Class Group mod m
Quadratic Residues

The Theory of Valued Fields

The Fundamental Concepts REgarding Valuations
Arithmetic in a Discrete Valued Field
The Completion of a Valued Field
The Completion of a Discrete Valued Field. The p-adic Number Fields
The Isomorphism Types of Complete Discrete Valued Fields with Perfect Residue Class Field
Prolongation of a Discrete Valuation to a Purely Transcendental Extension
Prolongation of a Discrete Valuation to a Purely Transcendental Extension
Prolongation of the Valuation of a Complete Field to a finite-Algebraic Extension
The Isomorphism Types of Complete Archimedean Valued Fields
The Structure of a Finite-Algebraic Extension of a Complete Discrete Valued Field
The Structure of the Multiplicative Group of a Complete Discrete Value Field with Perfect Residue Class field of Prime Characteristic
The Tamely Ramified Extension Types of a Complete Discrete Valued Field with Finite Residue Class Field of Characteristsic p
The Exponential Function, the Logarithm, and Powers in a Complete Non-Archimedean Valued Field of Characteristic 0
Prolongation of the Valuation of a Non-Complete Field to a Finite-Algebraic Extension

The Foundations of Arithletic in Algebraic Number Field

Relations Between the Complete System of Valuations and the Arithmetic of the Rational Number Field
Prolongation of the Complete System of Valuations to Finite-Algebraic Extension
The Prime Spots of an Algebraic Number Field and their Completions
Decomposition into Prime Divisors, Integrality, and Divisibility
Congruences
The Multiples of a Divisor
Differents and Discriminants
Quadratic Number Fields
Cyclomotic Fields
Units
The Class Number
Approximation Theorems and Estimates of a Discriminant
Index

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Helmut Hasse
Parution	05/02/2002
Nb. de pages	638
Format	15,5 x 23,5
Couverture	Broché
Poids	1000g
Intérieur	Noir et Blanc
EAN13	9783540427490