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.
Notations, Definitions, and Prerequisites
1. Principal ideal rings
2. Elements integral over a ring; elements algebraic over a field
Appendix: The field of complex numbers is algebraically closed
3. Noetherian rings and Dedekind rings
4. Ideal classes and the unit theorem
Appendix: The calculation of a volume
5. The splitting of prime ideals in an extension field
6. Galois extensions of number fields
A supplement, without proofs