This must be an easy question, but I am confused. We are given a complete discrete valuation field $K$ and its finite extension $L$.
I knew that in this setting it is true that $\mathcal O_K \subset \mathcal O_L$. My argument was that $\mathcal O_K$ consists of all elements of $K$ with valuation $\geq 0$ and $\mathcal O_L$ consists of all elements of $L$ with valuation $\geq 0,$ thus, any element of $\mathcal O_K$ being an element of $K\subset L$ and having valuation $\geq 0$ must lie inside $\mathcal O_L$. However, during this argument I imagined that valuation $\nu_L$ on $L$ is simply the unique continuation of valuation $\nu_K$ on $K$. Now I have huge doubts about that.
On the other hand, there is this notion of ramification index $e$, telling us that $\nu_K=\frac{1}{e} \nu_L.$ Hence, my assumption about $\nu_K$ and $\nu_L$ was wrong, so there must be other reason why $\mathcal O_K \subset \mathcal O_L$ and $\mathfrak{m}_K\subset \mathfrak{m}_L$. Of course, if $\nu_K(x)\geq 0$ then $\nu_L(x) \geq 0$ and previous argument still works, but for us even introducing ramification index and stating that $\nu_K=\frac{1}{e} \nu_L$ we at first must be sure that $\mathcal O_K \subset \mathcal O_L$. So, what is the explanation for this inclusion?
Thank you.