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Algebraic number fields

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 has finite dimension when considered as a vector space over Q.

The study of algebraic number fields, and these days 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
class field theory
Brauer group
Iwasawa theory
Dedekind zeta function.

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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Algebraic number fields".