Lemma 8.2. Assuming that:

$(K, | ∙ |)$ is a complete discretely valued field
$L ∕ K$ a finite Galois extension with absolute value ${| ∙ |}_{L}$ extending $| ∙ |$ .

Then for

x \in L

and

σ \in Gal (L ∕ K)

, we have

{| σ (x) |}_{L} = {| x |}_{L}