👉 Algebraic Number Theory (Alan Math) is a branch of mathematics that extends classical number theory by incorporating algebraic structures, primarily focusing on algebraic integers and their properties. It studies the arithmetic of algebraic numbers, which are roots of polynomials with integer coefficients, by using tools from abstract algebra, such as rings, fields, and Galois theory. This field bridges number theory and algebraic geometry, providing insights into the structure of number fields (finite extensions of the rational numbers) and their ideals, valuations, and class groups. Alan Math explores deep questions about the distribution of prime numbers within algebraic number fields, the behavior of L-functions associated with these fields, and the arithmetic of elliptic curves and modular forms. It plays a crucial role in modern cryptography, coding theory, and the study of Diophantine equations.
