# Ring of Witt Vectors and Tensor product of Fields

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.

-
It is not true in general that when $k$ is finite you will always have $k\subset k_1$ or $k_1\subset k$. – Kevin Ventullo Jul 12 '13 at 20:04
@KevinVentullo: Oops, sorry. I was being stupid. But the above result is still true in that case. I edited my question. Thanks anyway! – david Jul 12 '13 at 20:32
A family of special cases, including the ones of interest to you, of an earlier, more general question was addressed by @WilberdvanderKallen. – L Spice Jan 5 at 3:06