Proposition 1

Let $K$ be a field, $L$ an algebraic extension of $K$, $v'$ a valuation on $L$, $v$ its restriction to $K$ and $A$ and $A'$ the rings of $v$ and $v'$. Then $\Gamma_{v'}/\Gamma_v$ is a torsion group and $\kappa(A')$ is an algebraic extension of $\kappa(A)$.

Let $(L_\alpha)$ be the family of finite sub-extensions of $L$; let us write $\Gamma_\alpha = v'(L_\alpha^*)$. The group $\Gamma_{v'}$ is the union of the right directed family consisting of the $\Gamma_\alpha$; as the groups $\Gamma_\alpha/\Gamma_v$ are finite (Lemma 2), $\Gamma_{v'}/\Gamma_v$ is a torsion group. The argument is similar to prove that $\kappa(A')$ is an algebraic extension of $\kappa(A)$.