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Let $p > 2$ be a prime, and let $\textbf{F}_{p} = \textbf{Z}/p\textbf{Z}$. Let $k_{1}$ be a finite field over $\textbf{F}_{p}$, and let $k$ be a perfect field of characteristic $p$. Then we have ring isomorphism $k_{1} \otimes_{\textbf{F}_{p}} k \cong \oplus_{i=1}^{n} l_{i}$ where $l_{i}$ are finite extensions of $k$.

Question: How do we prove that $W(k_{1}) \otimes_{\textbf{Z}_{p}} W(k) \cong \oplus_{i=1}^{n} W(l_{i})$, where $W(k)$ denote the ring of Witt vectors of $k$?

Any suggestions or comments would be greatly appreciated.

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    $\begingroup$ It is not true in general that when $k$ is finite you will always have $k\subset k_1$ or $k_1\subset k$. $\endgroup$
    – Kevin Ventullo
    Commented Jul 12, 2013 at 20:04
  • $\begingroup$ @KevinVentullo: Oops, sorry. I was being stupid. But the above result is still true in that case. I edited my question. Thanks anyway! $\endgroup$
    – david
    Commented Jul 12, 2013 at 20:32
  • $\begingroup$ A family of special cases, including the ones of interest to you, of an earlier, more general question was addressed by @WilberdvanderKallen. $\endgroup$
    – LSpice
    Commented Jan 5, 2016 at 3:06

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