Synopsis
This book provides a clear and accessible introduction to modern algebraic number theory, with a special emphasis on class field theory. Drawing from courses and lectures delivered in seven countries, the text balances simplicity with depth, making advanced topics approachable with minimal reliance on heavy algebraic or analytic machinery. The book is structured into these distinct chapters; each tailored to a different stage of mathematical training: Chapter 1 introduces algebraic number fields, progressing at a steady pace with numerous examples — ideal for advanced undergraduates; Chapter 2 explores complete discrete valuation fields (local fields), a vital area of local algebraic number theory often underrepresented in standard textbooks but crucial in modern research; Chapter 3 develops abstract class field theory and its applications to both local and global fields, using Neukirch's axiomatic approach to derive fundamental theorems. Extensions to recent developments and generalizations are also discussed. The final chapter gathers a substantial collection of exercises, designed to test comprehension and guide further exploration. By offering the simplest known pathway to class field theory, this book fills a significant gap left by classic references published decades ago.
About the Author
Ivan Fesenko is a mathematician whose career spans several countries and decades, marked by pioneering contributions in number theory and proactive leadership in the mathematical sciences. His major areas of specialization include the core of algebraic number theory: class field theory, and its three generalizations. Fesenko introduced higher Haar measure and integration on higher local fields and higher adeles, and associated harmonic analysis. He launched a research program in higher adelic analysis and geometry, which extended the classical Iwasawa–Tate theory. This work involved the development of higher adelic integrals to study the zeta functions of elliptic surfaces using topological and measure-theoretic structures. He has studied Shinichi Mochizuki's IUT theory, co-authoring a 2022 paper that established effective abc-inequalities and offered a new proof of Fermat's Last Theorem as an application. In recent years, his interdisciplinary interests have expanded to encompass topics at the intersection of arithmetic geometry and quantum theory and computing, as well as new epidemic modelling. His advisory work in science policy has informed national strategies, contributing to a significant rise in funding for modern mathematical research.
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