Let $k$ be a field, with $F,k'$ field extensions of $k$. The ring $k' \otimes_k F$ is denoted by $F_{k'}$. In Borel's *Linear Algebraic Groups*, it is claimed (I believe erroneously) that "each of [$F_{k'}$'s] prime ideals is minimal." Indeed, by a result due to Grothendieck, the Krull dimension of $F_{k'}$ is the minimum of the transcendence degrees of $F/k, k'/k$. So there may be a tacit assumption that $F/k$ is algebraic, I am not sure. In any case, Borel claims results for three possibilities:

(a) $k'$ is separable algebraic over $k$.

Then $F_{k'}$ is reduced, but it may have more than one prime ideal.

(b) $k'$ is algebraic and purely inseparable over $k$.

Then $F_{k'}$ has a unique prime ideal (consisting of the nilpotent elements), but $F_{k'}$ need not be reduced.

(c) $k'$ is purely transcendental over $k$.

Then $F_{k'}$ is clearly an integral domain.

I was wondering if anyone might be able to provide references or proofs for any of the statements (a), (b), (c) as well as examples, e.g. an example where $F_{k'}$ has more than one prime ideal. Unfortunately, the reference Borel gives is an old algebraic geometry conference from the 1950s which I have had a lot of difficulty understanding.