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Librairie Eyrolles - Paris 5e

Classical Theory of Algebraic Numbers

Classical Theory of Algebraic Numbers

Paulo Ribenboim

681 pages, parution le 01/04/2001


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".


  • 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

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


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