What are the basic possibilities for a tensor product of two fields?

Looks like nfdc23 has explanations for (a),(b), and (c).

But: indeed, primes of $F_{k'}$ are not in general minimal if $F/k$ is not algebraic. Let $k'=k(x)$ and $F=k(y)$ be transcendental extensions of $k$.

Then $F_{k'}$ identifies with a subalgebra of the field $k(x,y)$, hence is an integral domain [assertion (c)] - i.e. $\{0\}$ is prime.

There is a homomorphism $\phi:F_{k'} = k' \otimes_k F \to k(t)$ - where $t$ is again transcendental - which on pure tensors is given by $f(x) \otimes g(y) \mapsto f(t) \cdot g(t)$. And then $x\otimes 1 - 1\otimes y$ is a non-zero element of $P = \ker \phi$, hence $P$ is a prime ideal which is not minimal.

For a), if $L$ is a finite Galois extension of $k$ with Galois group $G$, then

$$L \otimes_k L \cong \prod_{g \in G} L$$

has $|G|$ prime ideals. For b), if $k = \mathbb{F}_p(a)$ and $L = k[x]/(x^p - a)$, then

$$L \otimes_k L \cong L[x]/(x^p - a) \cong L[x]/(x - \sqrt[p]{a})^p$$

is not reduced. I haven't thought about c).