Algebraic Theory of Numbers: Translated from the French by Allan J. Silberger (Dover Books on Mathematics)

Marc Notes:
Originally published: Paris: Hermann; Boston: Houghton Mifflin Co., 1970.; Includes bibliographical references (p. [106]-107) and index.

Table of Contents:
Translator's IntroductionIntroductionNotations, Definitions, and Prerequisites1. Principal ideal rings2. Elements integral over a ring; elements algebraic over a fieldAppendix: The field of complex numbers is algebraically closed3. Noetherian rings and Dedekind rings4. Ideal classes and the unit theoremAppendix: The calculation of a volume5. The splitting of prime ideals in an extension field6. Galois extensions of number fieldsA supplement, without proofsExercisesBibliographyIndex

Publisher Marketing:
Algebraic number theory introduces students not only to new algebraic notions but also to related concepts: groups, rings, fields, ideals, quotient rings and quotient fields, homomorphisms and isomorphisms, modules, and vector spaces. Author Pierre Samuel notes that students benefit from their studies of algebraic number theory by encountering many concepts fundamental to other branches of mathematics -- algebraic geometry, in particular.
This book assumes a knowledge of basic algebra but supplements its teachings with brief, clear explanations of integrality, algebraic extensions of fields, Galois theory, Noetherian rings and modules, and rings of fractions. It covers the basics, starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number, and the more elementary theorems of Hilbert ramification theory. Numerous examples, applications, and exercises appear throughout the text.