Description
Book SynopsisThis self-contained and comprehensive textbook of algebraic number theory is useful for advanced undergraduate and graduate students of mathematics. The book discusses proofs of almost all basic significant theorems of algebraic number theory including Dedekind’s theorem on splitting of primes, Dirichlet’s unit theorem, Minkowski’s convex body theorem, Dedekind’s discriminant theorem, Hermite’s theorem on discriminant, Dirichlet’s class number formula, and Dirichlet’s theorem on primes in arithmetic progressions. A few research problems arising out of these results are mentioned together with the progress made in the direction of each problem.
Following the classical approach of Dedekind’s theory of ideals, the book aims at arousing the reader’s interest in the current research being held in the subject area. It not only proves basic results but pairs them with recent developments, making the book relevant and thought-provoking. Historical notes are given at various places. Featured with numerous related exercises and examples, this book is of significant value to students and researchers associated with the field. The book also is suitable for independent study. The only prerequisite is basic knowledge of abstract algebra and elementary number theory.
Trade Review“A Textbook of Algebraic Number Theory is intended to be used as a 2-term textbook for an algebraic number theory graduate course. … As a graduate course textbook, this would be an excellent resource. … I would definitely recommend this book for a graduate course following a thorough abstract algebra sequence. The topics covered are the foundations of the study of algebraic number theory.” (McKenzie West, MAA Reviews, October 9, 2023)
“This wonderful textbook will be of great help to everybody interested in algebraic number theory … . The book is an essence of a two-semester course on algebraic number theory held several times by the author to postgraduate students. … Readers will enjoy the presentation of the book together with interesting illustrations of historical notes.” (István Gaál, zbMATH 1500.11001, 2023)
Table of Contents1. Algebraic Integers, Norm and Trace.-2. Integral Basis and Discriminant.-3. Properties of the Ring of Algebraic Integers.- 4. Splitting of Rational Primes and Dedekind’s Theorem.-5. Dirichlet’s Unit Theorem.- 6. Prime Ideal Decomposition in Relative Extensions.- 7. Relative Discriminant and Dedekind’s Theorem on Ramified.- 8. Ideal Class Group.-9. Dirichlet’s Class Number Formula and its Applications.- 10. Simplified Class Number Formula for Cyclotomic, Quadratic Fields.
