Let $k$ be a field, $L$, $H$ extension fields of $k$, and $G=L\otimes_k H$. I wonder why (I want to know the proof but I can't find) the prime ideal of $G$ must be maximal, and its properties:

a) if $L$ is separable over $k$, then $G$ is reduced.

b) if $L$ is algebraic and purely inseparable over $k$, then $G$ has a unique prime ideal.