I came across this problem studying for exams, and it's been stumping me. Writing $$ \mathbb F_4 = \{0,1,\alpha, \alpha + 1\} = \mathbb F_2(\alpha) $$ gives that $\mathbb F_4 \otimes_{\mathbb F_2} \mathbb F_4$ is spanned by the elements $$ 1\otimes 1, 1\otimes \alpha, \alpha \otimes 1, \alpha \otimes \alpha.$$
I know how to compute in this ring, but I'm having trouble actually finding what the ideals are, much less the prime ones. Any help would be appreciated!
The extension $\mathbb F_2\hookrightarrow \mathbb F_4$ is Galois with Galois group $G=\{Id,\sigma\}$which means (or implies) that it diagonalizes itself through the $\mathbb F_4$-algebra isomorphism : $$j:\mathbb F_4 \otimes_{\mathbb F_2} \mathbb F_4\stackrel {\sim}{\to}\mathbb F_4\times \mathbb F_4:x\otimes y\mapsto (xy,x\sigma y)$$ The two prime ideals of $\mathbb F_4\times \mathbb F_4$ are $\mathfrak q_1=\{0\}\times \mathbb F_4$ and $\mathfrak q_2=\mathbb F_4\times \{0\}$.
Pulling these ideals back to $\mathbb F_4 \otimes_{\mathbb F_2}\mathbb F_4$ through $j^{-1}$ you find the the two required prime ideals: $$ \mathfrak p_1= j^{-1}\mathfrak q_1=\{x\alpha\otimes 1+x\otimes\alpha\vert x\in \mathbb F_4\}\subset \mathbb F_4 \otimes_{\mathbb F_2} \mathbb F_4 $$ $$\mathfrak p_2=j^{-1}\mathfrak q_2=\{y(\alpha +1)\otimes1+y\otimes\alpha\vert y\in \mathbb F_4\} \subset \mathbb F_4 \otimes_{\mathbb F_2} \mathbb F_4 $$