Algebraic Number Theory for Beginners

This accessible guide introduces algebraic number theory, focusing on unique prime factorization and its historical development in Algebraic Number Theory for Beginners.

The book Algebraic Number Theory for Beginners serves as an accessible introduction for undergraduate mathematics students and educators. It explores the intricacies of algebraic number theory through engaging problems derived from traditional number theory. By leveraging algebraic numbers, the text presents a thoughtful generalization of unique prime factorization, providing readers with a comprehensive understanding of the subject matter. The integration of historical context enriches the learning experience, offering insights into the development of key concepts.

At the heart of Algebraic Number Theory for Beginners lies the challenge of extending the notion of 'unique prime factorization' from ordinary integers to broader domains. This journey begins with solving polynomial equations in integers, which naturally leads to these more complex domains. However, the process may obscure the principle of unique prime factorization, necessitating the introduction of Dedekind's ideals to restore this crucial feature. The text emphasizes the importance of foundational concepts such as algebraic number fields and algebraic integers, alongside the supporting theories of rings, vector spaces, and modules.

The work of Emmy Noether is highlighted, as she encapsulated the properties of rings that facilitate unique prime factorization, now known as Dedekind rings. The author carefully develops these theories while tracing their historical roots, ensuring that each concept is grounded in its origins. The result is a self-contained, easily digestible text that is suitable for a one-semester course, making Algebraic Number Theory for Beginners an ideal choice for those new to the field.