Tensor product of algebraic closure with an algebraic extension of a field

Question

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?

Answer 1

In the simple case where $L/K$ is separable, and thus generated by a single algebraic element $\alpha$, we have $$L = K[x]/(p(x)),$$ where $p(x)$ is the minimal polynomial of $\alpha$ over $K$. Therefore $$L \otimes_K \bar{K} \cong \bar{K}[x]/(p(x)),$$ and since $\bar{K}$ is algebraically closed, $p(x)$ splits into powers of linear factors. Now the Chinese remainder theorem applies. If $L/K$ is not singly generated, the answer will be less convenient.

Answer 2

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...