Exercise 5, § 4

Let $K$ be a commutative field and let $L$ be a finite Galois extension of $K$ with Galois group $G$. Let $v_L$ be a valuation of $L$ with values in $\mathbf{Z}$ (Commutative Algebra, Chapter VI, § 3, no. 2) which is invariant under $G$. If $g \in G$, we write

Show that $v$ is the order function of a separated integral filtration $(G_n)$ on $G$ such that $G_0 = G$ and that the restriction of this filtration to $G_1$ is central (apply Exercise 3, taking $H$ to be the ideal of $v_L$). Show that $G_1$ is a $p$-group (resp. reduces to the identity element) if the residue field of $L$ is of characteristic $p > 0$ (resp. of characteristic zero); when the fields $L$ and $K$ have the same residue field, $G_1$ is the kernel of the homomorphism $\phi$ defined in Commutative Algebra, Chapter VI, § 8, Exercise 11 (b).