Classical Theory of Algebraic Numbers

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