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!)