# Basic example of extensions of residue fields.

Can anyone think of a simple example of the following: $B/A$ is an integral extension of DVRs with quotient fields $L$ and $K$ and residue fields $\bar{L}$ and $\bar{K}$, $L/K$ is finite dimensional and Galois, but $\bar{L}/ \bar{K}$ is not separable.

(The above situation is dealt with in the first chapter of Serre's Corps locaux and a concrete example would be very helpful. I imagine there's a very simple example, but I'm currently drawing a blank. Thanks!)

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Consider a function field galois extension such that a prime is totally ramified, then complete with respect to that prime. – Soarer Jan 5 '12 at 22:10
Consider $A= \mathbb{Z}_p[\zeta_p][t]_{(1-\zeta_p)}$ and $B = \mathbb{Z}_p[\zeta_p][t^{1/p}]_{(1-\zeta_p)}.$ Then $L := K(B)$ is the splitting field of $X^p -t$ over $K := K(A),$ but $\overline{L} = \mathbb{F}_p(t^{1/p})$ and $\overline{K} = \mathbb{F}_p(t).$ – jspecter Jan 5 '12 at 22:25
I think that does it. Thanks! – Brett Frankel Jan 5 '12 at 22:35