Einfuhrung in Die Elementare Und Analytische Theorie Der Algebraischen Zahlen Und Der Ideale

Presents an account of the theory of algebraic numbers and ideals leading up to Dedekind's theorem that every ideal of an algebraic number field is the unique product of prime ideals. This work studies Landau's generalization of the prime number theorem that was conjectured by Gauss and proven by Hadamard and Vallee Poussin in 1896.

The first part of this book provides a short and self-contained account of the theory of algebraic numbers and ideals leading up to Dedekind''s theorem that every ideal of an algebraic number field is the unique product of prime ideals. The author also proves the finiteness of class number as well as Dirichlet''s unit theorem. The second part studies Landau''s generalization of the prime number theorem that was conjectured by Gauss in the eighteenth century and proven by Hadamard and Vallee Poussin in 1896. Landau generalized this fundamental result to arbitrary number fields in 1903. A few months before the publication of this book in 1917, Hecke proved a ground-breaking result about the analytic continuation of Dedekind zeta functions.The inclusion of a proof of Hecke''s result in this book enabled Landau to reprove and sharpen his old prime ideal theorem as well as to present a variety of interesting corollaries. This book is based on lectures that the author gave in Berlin and Gottingen.