Algebraic number field

Finite extension of the rationals

In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle \mathbb {Q} } such that the field extension K / 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.

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