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.



question about  tensor of two fields 
For (a): a field extension $L/k$ is separable is $L$ is a separable algebra, that is, if $L\otimes K$ is a semisimple algebra for all field extensions $K/k$. In particular, if $L$ is separable over $k$ and $K$ is an extension of $k$, then $L\otimes K$ will have no nilpotent elements because it is semisimple.


For details, see for example, Pierce's beautiful book Associative algebras.


For (b), for which the argument is more elaborate, see Jacobson's *Lectures in Abstract Algebra*, vol. 3, where the result is part 3 of Theorem 21; he does without algebraicity, by the way.


