TITLE: Tensor product of algebraic closure with an algebraic extension of a field
QUESTION [6 upvotes]: Suppose $K$ be a field, $\bar{K}$ its algebraic closure, and $L$ some algebraic extension of $K$. I need to compute $\hbox{Spec}(L \otimes_K \bar{K})$. 

Is there some result from algebra which gives a nice description of $L \otimes_K \bar{K}$ which would help me compute the spectrum?

REPLY [7 votes]: The tensor product $A=L \otimes_K \bar{K}$ is an algebraic algebra over $\bar K$.
It is of Krull dimension zero i.e. every prime ideal of $ A$ is maximal.
I don't think I can say much more if you make no further hypothesis on $L$: the algebra $A$ can be infinite dimesional as a vector space over $\bar K$, it can be non reduced, have non trivial idempotents, etc...