|
In mathematics, the term global field refers to a field that is either: *an algebraic number field, i.e., a finite extension of Q, or *a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of F''q''(''T''), the field of rational functions in one variable over the finite field with ''q'' elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.〔 and 〕 ==Formal definitions== A ''global field'' is one of the following: ;An algebraic number field 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 of an algebraic curve over a finite field 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「global field」の詳細全文を読む スポンサード リンク
|