Algebraic number theory is the branch of number theory that deals with algebraic numbers. Historically,
algebraic number theory developed as a set of tools for solving problems in elementary
number theory, namely Diophantine equations
(i.e., equations whose solutions are integers or rational
numbers). Using algebraic number theory, some of these equations can be solved
by "lifting" from the field 
of rational numbers to an algebraic
extension 
of 
.
More recently, algebraic number theory has developed into the abstract study of algebraic numbers and number fields themselves, as well as their properties.
