In mathematics, a global field is one of two types of fields (the other one is local fields) that are characterized using valuations. There are two kinds of global fields:

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

We say that field ${\displaystyle K}$ is global field when there exists a set ${\displaystyle {\mathfrak {M}}}$ of places (equivalence classes of absolute values on ${\displaystyle K}$) such that:

Only two kind of fields satisfy the axiomatic definition:

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 an algebraic variety is the set of all rational functions on that variety. On an irreducible algebraic curve (i.e. a one-dimensional variety V) over a finite field, we define a rational function on an open affine subset U as the ratio of two polynomials in the affine coordinate ring of U, and a rational function on all of V consists of such local data that agree on the intersections of open affine subsets. 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.

There are a number of formal similarities between the two kinds of global fields. A field of either type has the property that all of its completions are locally compact fields (see local fields). Every field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index. In each case, one has the product formula for non-zero elements x:

where v varies over all valuations of the field.