Let $X$ be an integral Noetherian scheme of dimension $n$ over a field $k$ (arbitrary field).
The function field of $X$ is defined as $K(X):=\mathcal O_{X,\eta}$ where $\eta$ is the generic point of $X$.
A field $K\supset k $ is called an algebraic function field in $n$ variables over $k$ if the extension $K|k$ is finitely generated, regular (it means that $k$ is algebraically closed in $K$) and of transcendence degree $n$.
How can I prove that $K(X)$ is a an algebraic function field in $n$ variables over $k$? Also just a reference for the proof will be appreciated.
Important edit: Ok, this fact is well shown in literature but there is still a problem.
Usually, for an algebraic function field $K|k$ one requires that $k$ is algebraically closed in $K$ (namely $k$ is the field of constants). When is this true for the function field $K(X)|k$? Which are the conditions on $X$ and $k$?