Classical Theory of Algebraic Numbers

Paulo Ribenboim

681 pages, parution le 01/04/2001

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Résumé

Gauss created the theory of binary quadratic forms in "Disquisitiones Arithmeticae" and Kummer invented ideals and the theory of cyclotomic fields in his attempt to prove Fermat's Last Theorem. These were the starting points for the theory of algebraic numbers, developed in the classical papers of Dedekind, Dirichlet, Eisenstein, Hermite and many others. This theory, enriched with more recent contributions, is of basic importance in the study of diophantine equations and arithmetic algebraic geometry, including methods in cryptography. This book has a clear and thorough exposition of the classical theory of algebraic numbers, and contains a large number of exercises as well as worked out numerical examples. The Introduction is a recapitulation of results about principal ideal domains, unique factorization domains and commutative fields. Part One is devoted to residue classes and quadratic residues. In Part Two one finds the study of algebraic integers, ideals, units, class numbers, the theory of decomposition, inertia and ramification of ideals. Part Three is devoted to Kummer's theory of cyclomatic fields, and includes Bernoulli numbers and the proof of Fermat's Last Theorem for regular prime exponents. Finally, in Part Four, the emphasis is on analytical methods and it includes Dinchlet's Theorem on primes in arithmetic progressions, the theorem of Chebotarev and class number formulas. A careful study of this book will provide a solid background to the learning of more recent topics, as suggested at the end of the book. Paulo Ribenboim is the author of "13 Lectures on Fermat's Last Theorem", "Fermat's Last Theorem for Amateurs", "The Theory of Classical Valuations", "My Numbers, My Friends", as well as the widely known "The Little Book of Big Primes" and "The New Book of Prime Number Records".

Contents

Unique Factorization Domains, Ideals, Principal Ideal Domains
Commutative Fields
Residue Classes
Quadratic Residues
Algebraic Integers
Integral Basis, Discriminant
The Decomposition of Ideals
The Norm and Classes of Ideals
Estimates for the Discriminant
Units
Extension of Ideals
Algebraic Interlude
The Relative Trace, Norm, Discriminant and Different
The Decomposition of Prime Ideals in Galois Extensions
Complements and Miscellaneous Numerical Examples
Local Methods for Cyclotomic Fields
Bernoulli Numbers
Fermat's Last Theorem for Regular Prime Exponents
More on Cyclotomic Extensions
Characters and Gaussian Sums
Zeta-Functions and L-Series
The Dedekind Zeta- Function
Primes in Arithmetic Progressions
The Frobenius Automorphism and the Splitting of Prime Ideals
Class Number of Quadratic Fields
Class Number of Cyclotomic Fields
Miscellaneous Results about the Class Number of Cyclotomic Fields

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Paulo Ribenboim
Parution	01/04/2001
Nb. de pages	681
Format	16 x 24
Couverture	Relié
Poids	1161g
Intérieur	Noir et Blanc
EAN13	9780387950709
ISBN13	978-0-387-95070-9