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algebraic function field
In mathematics, an (algebraic) function field of ''n'' variables over the field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebraic function field of ''n'' variables over ''k'' may be defined as a finite field extension of the field ''k''(''x''1,...,''x''''n'') of rational functions in ''n'' variables over ''k''.
==Example==
As an example, in the polynomial ring ''k''() consider the ideal generated by the irreducible polynomial ''Y''2−''X''3 and form the field of fractions of the quotient ring ''k''()/(''Y''2−''X''3). This is a function field of one variable over ''k''; it can also be written as (with degree 2 over ) or as (with degree 3 over ). We see that the degree of an algebraic function field is not a well-defined notion.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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