Algebraic function field

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