Foundations of Algebraic Number Theory

Delve into the fascinating world of algebraic number theory, a field that extends classical number theory by using abstract algebra to solve problems involving integers. This course provides a structured introduction to its foundational ideas, laying the groundwork for deeper exploration. By the end of this course, you will be equipped with a robust understanding of number fields, algebraic integers, and the unique factorization properties of ideals, preparing you for further study in pure mathematics, cryptography, and related domains. What you'll learn: Understand the fundamental definitions of number fields, algebraic integers, and integral bases. Learn to identify and work with rings of integers within various number fields. Master the concept of ideals and their unique factorization into prime ideals in Dedekind domains. Apply properties of norms, traces, and discriminants to analyze algebraic number fields. Explore the significance of the class group and unit group in understanding the structure of number fields. The course begins with essential definitions and examples, progressively building concepts from number fields and algebraic integers to the theory of ideals and factorization. It culminates in an exploration of advanced structural properties. This course is designed for absolute beginners in algebraic number theory with a background in abstract algebra and basic number theory. No prior exposure to algebraic number theory is required. Start your journey into algebraic number theory and unlock a new perspective on numbers.