Explore the foundations of algebraic number theory and rings in this 48-minute math history lecture. Delve into the 19th-century development of extension fields and abstract rings, focusing on algebraic integers in number fields. Examine key examples like Gaussian integers, discussing their properties under arithmetic operations. Learn about the challenges of unique factorization in algebraic number rings and the solutions proposed by Kummer and Dedekind. Consider the foundational issues in current formulations and the potential need for new algebraic techniques. Gain insights into the historical evolution of these mathematical concepts and their importance in modern number theory.