Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2:=\mathbb F_p((Y))$.

Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{1}{p}$ and $|q|_1=1$ for all $q\in \mathbb F_p$. Analogously, equip $F_2$ with the $Y$-adic multiplicative absolute value $|\cdot|_2.$

Define $|\cdot|_{prod}$ on the ring $F:=F_1\otimes _{\mathbb F_p} F_2$ in the following way. If $c\in F$, then

$$|c|_{prod}:=\inf\left(\max_{i}\left(\prod|c_{1,i}|_1|c_{2,i}|_2 \right)\ | \ c=\sum^{n}_{i=1}c_{1,i}\otimes c_{2,i}\right)$$

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 $F$?

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

My guess is that such elements don't exist and I am able to show this on the subring $\mathbb F_p[X]\otimes _{\mathbb F_p} \mathbb F_p[Y]$. Using this, I thought I could extend the conclusion to $F$ using a density argument (using that $\mathbb F_p[X]\otimes _{\mathbb F_p} \mathbb F_p[Y]$ is dense in $\mathbb F_p[[X]]\otimes _{\mathbb F_p} \mathbb F_p[[Y]]$ ), but I failed.