Vol. 7, Issue 2, Part D (2025)

Algebraic number theory is a central area of modern mathematics that studies the arithmetic properties of algebraic integers and number fields. Building upon classical number theory, it provides a structural framework for understanding Diophantine equations, prime factorization, and higher reciprocity laws. This paper explores the foundational structures of algebraic number theory number fields, rings of integers, ideals, and class groups and their profound applications to cryptography, modular forms, and modern computational mathematics. By weaving historical developments with contemporary advances, the study highlights the enduring significance sof algebraic number theory as a bridge between pure mathematics and practical applications.