Let $F$ be an algebraically closed field of characteristic $p$ equipped with an absolute value $|\cdot|:F \rightarrow \mathbb{R}_{\ge 0}$ with respect to which $F$ is complete.

Define $|\cdot|_{prod}$ on the ring $F\otimes _{\mathbb F_p} F$ in the following way. If $c\in F\otimes _{\mathbb F_p} F$, then

$$|c|_{prod}:=\inf\left(\max_{1\le i\le n}\{|c_{1,i}||c_{2,i}| \}\ : \ c=\sum^{n}_{i=1}c_{1,i}\otimes c_{2,i}\right)$$

where the infimum is taken over all the possible ways to write $c$ as a sum of pure tensors. Does $|\cdot|_{prod}$ define a norm on $F\otimes _{\mathbb F_p} F$?

I am able to show that $|\cdot|_{prod}$ defines a semi-norm, which is submultiplicative and non-archimedean, but I am not able to find whether there does exist some $x\ne 0$ s.t. $|x|_{prod}=0$.

My guess is that such elements don't exist and I am also able to show that no non-zero pure tensor has absolute value zero, but I still can't show this for a general element of $F\otimes _{\mathbb F_p} F$.