Is there any counterexample of the statement that the residue field extension of a separable extension is also separable?

Question

Recently I'm reading Serre's Local Fields, and in page 22 (English version) he says that

The residue field extension $\overline L/\overline{K}$ is separable in each of the following cases (which cover most of the applications):

1. $\overline K$ is perfect;
2. The order of the inertia group $T$ is prime to the characteristic $p$ of the residue field $K$ (indeed, we have seen that the order of this group is divisible by $p^s$).

Here we assume that $A\subset B$ are Dedekind domains with fractional fields $K$ and $L$ and $L/K$ is a finite separable extension. Let $\mathfrak p$ be a prime of $A$, fix a $\mathfrak P$ which is a prime of $B$ and appears in the factorization of $\mathfrak p B $ in $B$, and $\overline K:=A/\mathfrak p, \overline L:=B/\mathfrak P$ are their residue fields.

My question is does there exist a counterexample of $A$ and $B$ making the extension $\overline L/\overline{K}$ not a separable extension. Furthermore, if we add the condition that $L/K$ is a Galois extension, does the counterexample still exist?

Answer 1

Suppose that $K$ has mixed characteristic $p$ (i.e., it has characteristic 0ドル$ but $\overline K$ has characteristic $p$), and $\overline K$ is not perfect. Upon replacing $K$ by a suitable totally ramified extension, we may assume that it has all $p$th roots of unity. Let $\bar t$ be a non-$p$th-power in $\overline K$, write $t$ for a lift of $\bar t$ to $A$, and consider $L = K(\sqrt[p]t)$.

• Fate Lie: Could you give an example of the field $K$ you mentioned?
• LSpice: @FateLie, take the fraction field of the Witt vectors of your favourite imperfect field.

Answer 2

LSpice gives a mixed characteristic example, it's also easy enough in same characteristic. Let $k$ have characteristic $p$, let $\overline{K} = k(u)$ and let $K= \overline{K}((t))$. Now let $L = K[x]/(x^p-tx-u)K[x]$. The derivative of this polynomial is $-t$, which is nonzero, so the extension of $K$ is separable, but the residue field extension is $\overline{K}(u^{1/p})$, so it is inseparable.

The Galois closure of $L$ is $L[t^{1/(p-1)}]$, so it is also separable over $L$ and thus over $K$.

• KConrad: A similar example that is Galois with degree $p$ is Example 3 in kconrad.math.uconn.edu/blurbs/gradnumthy/….