Home > Algebraic number field

In mathematics, an algebraic number field (or simply number field) is a finite field extension of the rational numbers Q. That is, it is a field which contains Q and has finite dimension when considered as a vector space over Q.

The study of algebraic number fields, and these days also of infinite algebraic extensions of the rational number field, is the central topic of algebraic number theory.

See in particular:

• quadratic field
• cyclotomic field
• Ideal class group
• Dirichlet's unit theorem
• local field
• global field
• abelian extension
• Kummer extension
• reciprocity law
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• Brauer groupIn mathematics, the Brauer group arose out of an attempt to classify division algebras over a given field K''. It is an abelian group with elements isomorphism classes of division algebras over K such that the center is exactly K''. The group is named for
• Iwasawa theoryField theory Algebraic number theory In number theory, Iwasawa theory is a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa as part of the theory of cyclotomic fields. Iwasawa's starting observation was that there are towers of fi
• Dedekind zeta functionIn mathematics, the Dedekind zeta function is a Dirichlet series defined for any algebraic number field K and denoted :ζ s where s is a complex variable. It is the infinite sum :Σ NI &minus s taken over all ideals I of the ring of integers O of.