Number field

In mathematics, an algebraic number field (or simply number field) is an extension fieldK{\displaystyle K} of the field of rational numbersQ{\displaystyle \mathbb {Q} } such that the field extensionK/Q{\displaystyle K/\mathbb {Q} } has finite degree (and hence is an algebraic field extension). Thus K{\displaystyle K} is a field that contains Q{\displaystyle \mathbb {Q} } and has finite dimension when considered as a vector space over Q{\displaystyle \mathbb {Q} }.The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods. DefinitionPrerequisitesThe notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication. A prominent example of a field is the field of rational numbers, commonly denoted