Function field

Remarks

Many authors also ask that $k$ be algebraically closed in $K$, and some ask $n \ge 1$. Example: the field of rational functions $k(x_1, \dots, x_n)$ is a function field in $n$ variables. Example: if $k$ has characteristic not $2$, the field $k(x)[y]/(y^2 - x^3 - 1)$ is a function field in one variable, with transcendence basis $\set{x}$; it is the function field of the curve $y^2 = x^3 + 1$. Non-example: the field $k(x_1, x_2, x_3, \dots)$ in infinitely many variables is not finitely generated over $k$. For an irreducible variety $V$ over $k$, the field $k(V)$ of rational functions on $V$ is a function field in $\dim V$ variables over $k$.