A Course in Algebraic Number Theory
This graduate-level text provides coverage for a one-semester course in algebraic number theory. It explores the general theory of factorization of ideals in Dedekind domains as well as the number field case. Detailed calculations illustrate the use of Kummer's theorem on lifting of prime ideals in extension fields.
The author provides sufficient details for students to navigate the intricate proofs of the Dirichlet unit theorem and the Minkowski bounds on element and ideal norms. Additional topics include the factorization of prime ideals in Galois extensions and local as well as global fields, including the Artin-Whaples approximation theorem and Hensel's lemma. The text concludes with three helpful appendixes. Geared toward mathematics majors, this course requires a background in graduate-level algebra and a familiarity with integral extensions and localization.
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absolute value AKLB setup algebraic integers algebraic number theory assume belongs calculate class number coefficients compute conjugate contains contradiction converges Corollary cyclotomic extension decomposition group Dedekind domain deﬁne deﬁnition Definitions and Comments discrete valuation divides embeddings equivalent ﬁeld field discriminant ﬁnd ﬁnite ﬁrst fraction field fractional ideal free Z-module Frobenius automorphism fundamental domain geometry hence homomorphism hypothesis ideal class group integral basis integral domain integral ideal integrally closed isomorphic Lemma Let linear local ring matrix maximal ideal minimal polynomial Minkowski bound mod Q monic multiplication Noetherian nonarchimedean nontrivial nonzero element nonzero fractional ideal nonzero ideal nonzero prime ideal number field positive integer prime factorization primitive principal ideal Problems For Section Proof proper ideal Proposition Let ramify rational numbers relatively prime result follows ring of algebraic root of unity Show subgroup subring trace unique factorization vector space