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
• additive polynomial
• 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.