TITLE: Spec of tensor product of fields
QUESTION [7 upvotes]: Suppose $K/k$ is a finite separable extension of degree $n$. How to show that there exists a finite separable extension $k'/k$ such that $\operatorname{Spec}(K \otimes_k  k') $ consists of $n$ points?

(It is in Gortz and Wedhorn, Algebraic Geometry I, Exercise 4.16)

REPLY [15 votes]: Probably it is wrong (I'm not familiar with field theory) but I think the following observations could be useful: 

A finite separable extension of $k$ has a primitive element. Call $p(x)$ its minimal polynomial: it has degree $n$ and it is separable.
$K=\frac{k[x]}{(p(x))}$
Consider $k'$ as the splitting field of $p(x)$, it is a separable extension of $k$, because $p(x)$ is separable and, if $p(x)= (x-\alpha_1)\cdots (x-\alpha_n)$, we have the following equalities: $$K \otimes_k k'=\frac{k[x]}{(p(x))}\otimes_k k'= \frac{k'[x]}{(p(x))}= \frac{k'[x]}{(x-\alpha_1)}\times \dots \times \frac{k'[x]}{(x-\alpha_n)}=(k')^n$$