I try to understand the index of ramification on the p-adic analysis. More precisely, I want to check the following statement via a specific example.
Let $\mathbb{Q}_p$(p:odd prime) be the completion of $\mathbb{Q}$ and $L$ be an extension field of $\mathbb{Q}_p$, let $\pi \in L$ be any element such that $ord_p \pi = (1/e)$. Then $n=e \cdot f$ (Where $n=[L:\mathbb{Q}_p]$ , and $f$ is the degree of the residue field $A/M$ over $\mathbb{F}_p$. [page66, Koblitz p-adic Numbers, p-adic Analysis, and Zeta-Functions]
Here is the specific example. (More on extensions of valuations and ramification, 19~20 pages, the PDF). According to the PDF, $L/K$ be a finite extension, with K complete with respect to a discrete absolute value $|\cdot|$. $\mathcal{O}_K$ is a valuation ring of K , with maximal ideal $\mathfrak{p}_K=(\pi_K)$ (and so is the case of $\mathcal{O}_L,$ with $\mathfrak{p}_L=(\pi_L)$)
Example) $K=\mathbb{Q}_p$ , $L=\mathbb{Q}_p[\sqrt{u}]$ (where p is odd prime number and $u \in \mathbb{Z}^{\times}$) then each reside field corresponds to $k=\mathbb{F}_p, ~l=\mathbb{F}_p[\sqrt{\bar{u}}] $. Thus, $f=[l:k]=2$. However, since $p \mathcal{O}_L$ is a prime of $\mathcal{O}_L$. So $\pi_L=p=\pi_k$ and $e=1$ .
Obviously, the statement holds. But here is my question: how to yield the value $e$ and $f$ ?
My thought or guess is here. First, I constructed the valuation ring. By the equivalence realtion of valuation ring, $\mathcal{O}_K=\left \{ x \in K^{\times} : |x|_p \ge 0 \right \} \cup \left \{ 0\right \} $, is valuation ring where $|\cdot|_p$ is a p-adic norm.(clearly, by the definition of p-adic norm, a valuation of K with values in $G=\mathbb{R}$ , $\nu=|\cdot|_p : K^{\times} \to \mathbb{R}$ satisfies the two aximos.). Since valuation ring is a local ring(i.e a unique maximal ideal) and PID as well, clearly exists a unique maximal ideal $\mathfrak{p}_K=(\pi_K)$. Thus, a residue field $k$ is $\mathcal{O}_K/(\pi_K)$. However, I don't know why $\mathcal{O}_K/(\pi_K)$ is isomorphic to $\mathbb{F}_p$.(Of course, when thinking first isomorphism theorem, I vaguely imagine a homomorphism $\varphi: \mathcal{O}_L \to \mathbb{F}_p$ and natural mapping $\pi: \mathcal{O}_K \to k $, but it is also difficult for me because I cannot specifiy the maximal ideal $(\pi_K)$(Furthermore, I just guess valuation ring $\mathcal{O}_K$.) And since I cannot spcifiy the maximal ideal, I do not know why $e=1$
(Of course, since I cannot well understand this process, I cannot imagine $\mathcal{O}_L/(\pi_L)$ is isomorphic to $\mathbb{F}_p[\sqrt{\bar{u}}]$ )