Algebraic number theory, also known as algebraic theory of numbers, is a branch of number theory that involves the study of algebraic numbers and their properties. It is an interesting and complex field that draws upon concepts from algebra, geometry, and analysis. Synonyms for algebraic number theory include arithmetic geometry, pure algebraic number theory, and modern number theory. One can also associate it with algebraic geometry, which is another field closely related to this branch of study. Algebraic number theory is used in various applications such as cryptography, coding theory, and computational number theory.