Extension of a discrete valuation on a complete field

Question

Let $K$ be a complete field w.r.t discrete absolute value $|\cdot|_K$,

$\mathcal O_K=\{x\in K:|x|_K\leq 1\}$.

$L$ is an extension field with $[L:K]< \infty$ and let $\mathcal O_L$ be the integral closure of $\mathcal O_K$ in $L$.

Also let $\pi$ be the generator of the unique maximal ideal of the DVR $\mathcal O_K$.

If $\pi \mathcal O_L=\mathcal P^e$, with $\mathcal P$ the unique non zero prime in $\mathcal O_L$, then is it true that the absolute value $|\cdot|_{\mathcal P}$ on $L$ extends $|\cdot|_K$ on $K$?

Many thanks for your help.

Answer 1

If depends on how you define $|\pi|_K$ and $|\mathcal{P}|_L$. But if you define $|x|_L = \sqrt[{[L/K]}]{|N_{L/K}(x)|}$ for all $x\in L$ where $N_{L/K}(x)$ is the norm (*) of $x$ over $K$, then yes. Have you already heard about ramification indexes ?

(*) Recall the $N_{L/K}(x)$ is the product of all conjugates of $x$ in $L$ over $K$, which is the same as the determinant of the $K$-endomorphism $y\mapsto xy$ of $L$.