Algebraic function field
In mathematics, an algebraic function field (often abbreviated as function field) of n {\displaystyle n} variables over a field k {\displaystyle k} is a finitely generated field extension K / k {\displaystyle K/k} which has transcendence degree n {\displaystyle n} over k {\displaystyle k} . Equivalently, an algebraic function field of n {\displaystyle n} variables over k {\displaystyle k} may be defined as a finite field extension of the field K = k ( x 1 , … , x n ) {\displaystyle K=k(x_{1},\dots ,x_{n})} of rational functions in n {\displaystyle n} variables over k {\displaystyle k} .