Editing Global field (section)

==Formal definitions==
{{main article|Algebraic number field|Function field of an algebraic variety}}
A ''global field'' is one of the following:
;An algebraic number field
An algebraic number field ''F'' is a finite (and hence [[algebraic extension|algebraic]]) [[field extension]] of the [[field (mathematics)|field]] of [[rational number]]s '''Q'''.  Thus ''F'' is a field that contains '''Q''' and has finite [[Hamel dimension|dimension]] when considered as a [[vector space]] over '''Q'''.

;The function field of an irreducible algebraic curve over a finite field
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 say that a rational function on an open affine subset ''U'' is defined as the ratio of two polynomials in the [[affine variety|affine coordinate ring]] of ''U'', and that a rational function on all of ''V'' consists of such local data that 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.