A property of tamely ramified extensions

Question

Suppose that $L/K$ is a tamely ramified extension of complete fields.

By computing some examples (quadratic fields, and cyclotomic fields $\mathbb Q_p(\zeta_p)/\mathbb Q_p$) I noticed that

$$ v(\Delta_{L/K})=[L:K]-f_{L/K}, $$

where $v$ is the valuation of $K$, $\Delta_{L/K}$ the discriminant of the extension, and $f_{L/K}$ the degree of the residue field.

I was wondering, is this always true for tame extensions? If yes, I would be interested in a proof or a reference. If not, is there any counterexample?

Answer 1

In a tamely ramified extension the different is $D_{L/K}=(\pi_L)^{e_{L/K}-1}$ (see e.g. Neukirch's ANT Theorem III 2.6), so $\Delta_{L/K} = N_{L/K}(D_{L/K}) = (\pi_K)^{f_{L/K}(e_{L/K}-1)}$. Since $f_{L/K}e_{L/K}=[L:K]$, we get $v(\Delta_{L/K}) = [L:K]- f_{L/K}$