Class Field Theory: An Introduction to Advanced Number Theory

Transitioning from basic algebraic number theory to the deep structures of class field theory can feel like an overwhelming leap. This text-based course bridges that gap by offering a highly structured, step-by-step guide to the symmetries of abelian extensions of global and local fields. Through clear written explanations, detailed proof breakdowns, and structured mathematical derivations, you will develop an intuitive grasp of class field theory's core machinery, transitioning from calculating simple quadratic cases to understanding the profound connections between Galois groups and arithmetic properties. What you'll learn: Understand the foundational definitions and historical context of class field theory; Analyze quadratic extensions and local fields using Hilbert symbols; Apply Galois cohomology to construct and interpret class field proofs; Explore modern connections between Galois representations and modular forms; Practice solving complex algebraic problems through guided written exercises; Verify theoretical results using modern computational mathematics tools. The course begins with essential terminology, local fields, and the quadratic case before gradually introducing Galois cohomology and global class field theory. You will conclude by exploring modern arithmetic geometry topics, including an introduction to modular forms. This course is designed for mathematics students, researchers, and self-learners who have a basic background in abstract algebra and elementary number theory and want a clear, text-focused pathway to advanced study. Start reading today to unlock the deep structural secrets of algebraic number fields.