Algebraic number field | StudentB Wikipedia Search

Algebraic structure → Ring theory Ring theory
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Commutative algebra Commutative rings • Integral domain • Integrally closed domain • GCD domain • Unique factorization domain • Principal ideal domain • Euclidean domain • Field • Finite field • Polynomial ring • Formal power series ring Algebraic number theory • Algebraic number field • Integers modulo $n$ • Ring of integers • p-adic integers $\mathbb {Z} _{p}$ • p-adic numbers $\mathbb {Q} _{p}$ • Prüfer p-ring $\mathbb {Z} (p^{\infty })$
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In mathematics, an algebraic number field (or simply number field) is an extension field $K$ of the field of rational numbers $\mathbb {Q}$ such that the field extension $K/\mathbb {Q}$ has finite degree (and hence is an algebraic field extension). Thus $K$ is a field that contains $\mathbb {Q}$ and has finite dimension when considered as a vector space over $\mathbb {Q}$ .

The study of algebraic number fields, and, more generally, 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.