Exercise 7, § 8

Let $K$ be a field, $v$ a valuation on $K$, $A$ the ring of $v$ and $m$ the ideal of $v$. Suppose that $A$ is Henselian; let $L$ be a finite extension of $K$ of degree $n$; let $v'$ be the unique valuation on $L$ extending $v$ (Exercise 6 (e)), $A'$ its ring and $m'$ its ideal. Let $x$ be any element of $A'$; show that the degree over $A/m$ of the class $\bar{x}$ of $x$ in $A'/m'$ is a divisor of $n$ and that the order of the class of $v'(x)$ in $\Gamma_{v'}/\Gamma_v$ is a divisor of $n$ (consider the minimal polynomial of $x$ over $K$).