A Textbook of Algebraic Number Theory

This 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.

Serves as a comprehensive textbook of algebraic number theory Discusses proofs of almost all significant basic theorems of algebraic number theory Presents numerous solved examples, exercises and problems in every chapter

Auteur

Sudesh Kaur Khanduja is Emeritus Professor at the Department of Mathematics, Panjab University, India, and INSA Senior Scientist at the Indian Institute of Science Education and Research (IISER) Mohali, India. A PhD and master's degree from Panjab University, India, her primary research interests are in algebraic number theory and valuation theory. With over 40 years of teaching experience at Panjab University and IISER Mohali, she has guided 12 PhD students and published over 85 research papers in reputed international journals.

A fellow of The World Academy Sciences, the Indian Academy of Sciences, the National Academy of Sciences, and the Indian National Science Academy (INSA), Prof. Khanduja was awarded the Professor V.V. Narlikar Memorial Lecture Award of INSA in 2015. She also has participated in various programs promoting the cause of mathematics at Panjab University. She has visited and delivered lectures at various universities including Ohio State University, Columbus; University of Missouri, Columbia; University of Michigan, Ann Arbor; University of Saskatchewan, Canada; Nihon University, Japan; State University of Campinas, Brazil, and University of Konstanz, Germany.

Résumé
"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)

Contenu

1. 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.