The following is the proposition (1.4) of Mumford's book Algebraic Geometry
If $\mathbb C$ has infinite transcendence degree over $k$, then every variety has a $k$-generic point.
In the proof
Then $L$ is an extension field of $k$ of finite transcendence degree. But any such field is isomorphic to a subfield of $\mathbb C$. i.e. there exists a monomorphism $\phi : L \rightarrow {\mathbb C}$.
Is any extension field of $k$ of finite transcendence degree isomorphic to a subfield of $\mathbb C$ ? If so, how do you prove it.