Corollary 3

Suppose that $L$ is a finite extension of $K$. For $v'$ to be discrete, it is necessary and sufficient that $v$ be discrete. If $v'$ is discrete, $\Gamma_{v'}$ is isomorphic to a non-zero subgroup of $\mathbf{Z}$ (Corollary 2) and hence to $\mathbf{Z}$. Conversely, if $v$ is discrete, $\Gamma_v$ is isomorphic to $\mathbf{Z}$ and $\Gamma_{v'}/\Gamma_v$ is a finite group (Lemma 2); hence $\Gamma_{v'}$ is a finitely generated commutative group of rank 1 and torsion-free; consequently it is isomorphic to $\mathbf{Z}$.