Classical Theory of Algebraic Numbers [Paperback]

The exposition of the classical theory of algebraic numbers is clear and thorough, and there is a large number of exercises as well as worked out numerical examples. A careful study of this book will provide a solid background to the learning of more recent topics.

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 cyclotomic 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 Dirichlet?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.* 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 * lı