A Gentle Course in Local Class Field Theory
Local class field theory is a fundamental and fascinating branch of algebraic number theory. It provides a deep understanding of the arithmetic of local fields and their extensions. A Gentle Course in Local Class Field Theory, written by Jean-Pierre Serre, is a highly regarded textbook that offers a comprehensive introduction to this subject. This article aims to provide an overview of the key concepts and objectives of this course, highlighting its significance in the field of algebraic number theory.
In the first chapter, the course introduces the basic notions of local fields, including their ring of integers, valuation, and absolute value. The reader is then guided through the construction of the local class group, which plays a crucial role in understanding the arithmetic of local fields. This chapter lays the foundation for the subsequent discussions on class field theory.
The second chapter delves into the theory of abelian extensions of local fields. It introduces the concept of a norm and shows how it can be used to construct abelian extensions. The reader is then introduced to the local class field theory, which states that every finite abelian extension of a local field is a subfield of its Hilbert class field. This chapter provides a detailed proof of this fundamental result, using the theory of norm and trace.
The third chapter explores the non-abelian extensions of local fields. It introduces the concept of a unit group and shows how it can be used to construct non-abelian extensions. The reader is then introduced to the theory of local class field theory for non-abelian extensions, which states that every finite extension of a local field is a subfield of its Hilbert class field. This chapter provides a detailed proof of this result, using the theory of norm and trace.
The fourth chapter discusses the relationship between local class field theory and the theory of Galois representations. It introduces the concept of a Galois representation and shows how it can be used to study the arithmetic of local fields. The reader is then guided through the construction of the local class field theory using Galois representations, providing a deeper understanding of the subject.
The fifth chapter presents applications of local class field theory to other areas of mathematics. It discusses the connection between local class field theory and the theory of zeta functions, as well as its applications in number theory and algebraic geometry. This chapter highlights the importance of local class field theory in understanding the arithmetic of number fields and their extensions.
In conclusion, A Gentle Course in Local Class Field Theory is an essential resource for anyone interested in learning about this fascinating subject. The course provides a clear and concise introduction to the key concepts and results of local class field theory, making it accessible to both students and researchers. By following the course, readers will gain a deeper understanding of the arithmetic of local fields and their extensions, and their applications in other areas of mathematics.
