Global field

In mathematics, the term global field refers to a field that is either:

An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.

Formal definitions

A global field is one of the following:

An algebraic number field F is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q.

A function field of a variety is the set of all rational functions on that variety. On an algebraic curve (i.e. a one-dimensional variety V) over a finite field, we say that a rational function on an open affine subset U is defined as the ratio of two polynomials in the affine coordinate ring of U, and that a rational function on all of V consists of such local data which agree on the intersections of open affines. This technically defines the rational functions on V to be the field of fractions of the affine coordinate ring of any open affine subset, since all such subsets are dense.