Consider the $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathbb{Z}_p$ and residue field $\mathbb{F}_p$. Let us take a finite extension $K$ of $\mathbb{Q}_p$, with ring of integers $\mathcal{O}_K$ and $k$ be the residue field of $K$. Let $\pi \in \mathcal{O}_K$ be a uniformizer. Consider the ring of power series $\mathfrak{S}\mathrel{:=}W(k)[[u]]$, where $W(k)$ is the ring of Witt vectors on the residue field $k$. Let $\mathcal{O}:=\widehat{\mathfrak{S}[\frac{1}{u}]}$ be the $p$-adic completion of $\mathfrak{S}[\frac{1}{u}]$. Also let $F$ be finite field.
Then $\mathcal{O} \otimes_{\mathbb{Z}_p} F$ decomposes as a product of fields.
Question: Is $\mathcal{O} \otimes_{\mathbb{Z}_p} F$ a finite product of fields ? Or an arbitrary product of fields ?
For example, $W(k) \otimes_{\mathbb{Z}_p} W(k)$ is a finite ($[k: \mathbb{F}_p]$ copies) direct product of $W(k)$.
But I couldn't conclude the question.
Any hints please
Correct me if I'm wrong but $W(k)= O_L=\Bbb{Z}_p[\zeta_{q-1}]$ where $L=\Bbb{Q}_p(\zeta_{q-1}),q=|k|$ is the largest unramified extension $\subset K$ $$\mathfrak{S}[\frac{1}{u}]=O_L[[u]][u^{-1}]=\{ \sum_{n\ge -N} a_n u^n, a_n\in O_L\}$$ whose $p$-adic completion is $$\mathcal{O} = \{ \sum_{n=-\infty}^\infty a_n u^n, a_n\in O_L, \lim_{n\to -\infty} v_p(a_n)=\infty\}$$ Note that $\mathcal{O}/(p)=\Bbb{F}_q((u))$.
For a finite field $F$, when $char(F)\ne p$ then $\mathcal{O}\otimes_{\Bbb{Z}_p}F=\mathcal{O}/(char(F))\otimes_{\Bbb{Z}_p}F=0$,
when $char(F)=p,F=\Bbb{F}_p[x]/(f(x))$ then $$\mathcal{O}\otimes_{\Bbb{Z}_p}F=\mathcal{O}/(p)\otimes_{\Bbb{F}_p}\Bbb{F}_p[x]/(f(x))$$ $$=\Bbb{F}_q((u))\otimes_{\Bbb{F}_p} \Bbb{F}_p[x]/(f(x))=\Bbb{F}_q((u))[x]/(f(x))$$ $$=\Bbb{F}_q((u))[x]/(\prod_j f_j(x))=\prod_j\Bbb{F}_q((u))[x]/(f_j(x))=\prod_j \Bbb{F}_{q^{\deg(f_j)}}((u))$$