Classical Theory of Algebraic Numbers

Description

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. Front Matter....Pages i-xxiv Front Matter....Pages 1-3 Unique Factorization Domains, Ideals, and Principal Ideal Domains....Pages 5-11 Commutative Fields....Pages 13-31 Front Matter....Pages 33-35 Residue Classes....Pages 37-59 Quadratic Residues....Pages 61-81 Front Matter....Pages 83-83 Algebraic Integers....Pages 85-105 Integral Basis, Discriminant....Pages 107-121 The Decomposition of Ideals....Pages 123-139 The Norm and Classes of Ideals....Pages 141-151 Estimates for the Discriminant....Pages 153-166 Units....Pages 167-187 Extension of Ideals....Pages 189-205 Algebraic Interlude....Pages 207-232 The Relative Trace, Norm, Discriminant, and Different....Pages 233-257 The Decomposition of Prime Ideals in Galois Extensions....Pages 259-272 The Fundamental Theorem of Abelian Extensions....Pages 273-289 Complements and Miscellaneous Numerical Examples....Pages 291-336 Front Matter....Pages 337-337 Local Methods for Cyclotomic Fields....Pages 339-366 Bernoulli Numbers....Pages 367-397 Fermat’s Last Theorem for Regular Prime Exponents....Pages 399-427 More on Cyclotomic Extensions....Pages 429-460 Front Matter....Pages 461-461 Characters and Gaussian Sums....Pages 463-485 Zeta-Functions and L -Series....Pages 487-503 The Dedekind Zeta-Function....Pages 505-521 Primes in Arithmetic Progressions....Pages 523-542 The Frobenius Automorphism and the Splitting of Prime Ideals....Pages 543-566 Class Numbers of Quadratic Fields....Pages 567-593 Class Number of Cyclotomic Fields....Pages 595-616 Miscellaneous Results About the Class Number of Quadratic Fields....Pages 617-637 Miscellaneous Results About the Class Number of Cyclotomic Fields....Pages 639-663 Back Matter....Pages 665-682