Editing Algebraic number theory (section)

{{Short description|Branch of number theory}}
{{Ring theory sidebar}}
[[Image:Disqvisitiones-800.jpg|thumb|Title page of the first edition of [[Disquisitiones Arithmeticae]], one of the founding works of modern algebraic number theory]]

'''Algebraic number theory''' is a branch of [[number theory]] that uses the techniques of [[abstract algebra]] to study the [[integers]], [[rational numbers]], and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as [[algebraic number field]]s and their [[rings of integers]], [[finite field]]s, and [[Algebraic function field|function field]]s. These properties, such as whether a [[ring (mathematics)|ring]] admits [[unique factorization]], the behavior of [[ideal (ring theory)|ideal]]s, and the [[Galois group]]s of [[field (mathematics)|field]]s, can resolve questions of primary importance in number theory, like the existence of solutions to [[Diophantine equation]]s.