18 786 Topics In Algebraic Number Theory Spring 2006

Unlock the beauty of modern mathematics by exploring the deep relationships between algebra and prime numbers. This course offers a clear, structured introduction to algebraic number theory, making complex abstract concepts accessible through step-by-step written explanations. Key algebraic structures often defy standard arithmetic rules. In this course, you will transition from basic algebra to analyzing advanced number fields, learning why unique factorization fails and how to restore it using ideals. By reading through detailed proofs, conceptual breakdowns, and practical computational examples, you will build a strong intuitive grasp of how modern number theory operates. What you'll learn: - Understand the fundamentals of number fields, rings of integers, and factorization - Calculate class numbers and apply Dirichlet's unit theorem to analyze algebraic structures - Explore local fields, valuations, and the mechanics of ramification - Analyze decomposition and inertia groups within Galois extensions - Apply modern computer algebra concepts using SageMath and Python to verify number-theoretic properties - Grasp the foundational elements of class field theory and analytic methods The course begins with essential terminology and foundational definitions, building your knowledge from basic ring theory up to advanced topics like local fields. Each concept is paired with clear written explanations of computational algorithms to help you practice the material. This course is designed for undergraduate students, educators, and math enthusiasts seeking a structured, beginner-friendly entry point into advanced algebra. No prior advanced number theory background is required. Start your journey into the heart of modern algebra and discover the power of algebraic number theory today.