Algebraic Number Theory
Algebraic Number Theory is a branch of number theory that studies algebraic numbers, which are complex numbers that are roots of polynomials with integer coefficients. It involves applying techniques from abstract algebra to problems in number theory. The subject can be divided into classical algebraic number theory, dealing with questions about unique factorization and ideal classes, and computational algebraic number theory, which focuses on algorithmic aspects and explicit calculations. Algebraic Number Theory has deep connections with other areas of mathematics such as geometry, topology, and representation theory. It also has applications in coding theory and cryptography, making it a vital area of study in the broader context of Science and Mathematics, particularly within Analytic Number Theory.