Introduction to Algebraic Number Theory: Foundations of Number Fields

Why do classical rules of arithmetic fail in some number systems, and how can we restore them? Algebraic number theory provides the elegant answers to these questions by extending the concept of integers to wider algebraic contexts. This course guides you from foundational algebraic definitions to advanced arithmetic theorems, giving you a rigorous understanding of how unique factorization behaves in general number fields. You will learn to analyze rings of integers, compute ideal class groups, and work with local fields. What you will learn: Understand the structure of Dedekind domains and the unique factorization of prime ideals; Analyze number fields, their rings of integers, and the splitting of primes; Calculate discriminants and utilize lattice methods to prove the finiteness of the class number; Apply Dirichlet's Unit Theorem to determine the structure of unit groups; Explore local fields, p-adic numbers, and the mechanics of ramification; Practice computational techniques used in modern mathematical software to analyze algebraic structures. The course starts with basic terminology and foundational ring theory before progressing to geometric methods and local-global principles. You will work through detailed explanations and conceptual exercises designed to solidify your mathematical intuition. This text-only course is designed for math enthusiasts and students who have a basic background in abstract algebra but are completely new to algebraic number theory. Begin your journey into the elegant world of algebraic numbers today.