Class Field Theory

Class Field Theory compiles lectures by Emil Artin from his 1951-1952 course at Princeton University. The notes present the foundations and key theorems of class field theory, a branch of algebraic number theory that studies abelian extensions of number fields. The text moves from basic concepts to advanced applications through a sequence of proofs and examples. Topics include local class field theory, the reciprocity law, and the theory of simple algebras in relation to class fields. The presentation emphasizes clarity and mathematical rigor while keeping notation consistent throughout. The progression builds systematically from fundamental elements to complex theoretical structures. These lectures represent a watershed moment in the development of modern algebra and number theory. The work continues to influence mathematical research and serves as a model for mathematical exposition.

Readers note this book presents advanced mathematical content in a clear, precise manner. The text compiles Artin's 1951-1952 lectures at Princeton into a concise treatment of class field theory. Liked: - Logical progression of concepts - Rigorous but readable proofs - Effective use of examples - Compact length (100 pages) Disliked: - Requires extensive background in algebra and number theory - Some sections feel rushed - Limited exercises/problems - Dated notation in places From Goodreads (3.8/5 from 5 ratings): "Direct and to-the-point presentation" - Mathematics PhD student "Too terse for self-study" - Graduate student From Amazon (4/5 from 3 ratings): "A classic treatment, but not for beginners" "Best read alongside more modern texts" The book remains in print but receives few new reviews, as most readers now use updated textbooks that build on Artin's work.

Algebraic Number Theory by Serge Lang The text develops number theory and class field theory from first principles with a focus on local and global fields.

Local Fields by Jean-Pierre Serre The book presents the theory of local fields, their extensions, and Galois cohomology as foundational elements of class field theory.

A Classical Introduction to Modern Number Theory by Kenneth Ireland, Michael Rosen The work connects classical number theory to modern algebraic techniques through field extensions and reciprocity laws.

Algebraic Number Fields by Gerald J. Janusz The text builds from basic algebraic number theory to the construction of class fields and proof of existence theorems.

Cohomology of Number Fields by Jürgen Neukirch, Alexander Schmidt, Kay Wingberg The book presents class field theory through the lens of Galois cohomology and etale cohomology of number fields.

🔸 Emil Artin developed much of class field theory during the 1920s and 1930s, revolutionizing how mathematicians understood number theory and algebraic structures. His book crystallized these groundbreaking ideas for future generations. 🔸 The theory presented in the book provides a deep connection between Galois theory and the arithmetic of number fields, explaining a centuries-old question about which numbers can be expressed as sums of squares. 🔸 Though published in English in 1967, the book originated from Artin's lectures at Princeton in 1951-52, where his students' carefully preserved notes formed the basis of this influential text. 🔸 Class field theory helped solve part of Hilbert's 9th problem from his famous list of 23 mathematical problems presented in 1900, specifically regarding the most general law of reciprocity in number fields. 🔸 Artin fled Nazi Germany in 1937, and despite facing significant personal challenges, continued his mathematical work in America, where he influenced an entire generation of algebraists through his teaching and this seminal work.