Completion of a number field w.r.t. $\mathfrak{p}$-adic valuation

Question

Let $F$ be a number field and let $\mathfrak{p}\in \mathsf{Spec} \: \mathcal{O}_F$. We have a non Archimedean valuation $\nu_\mathfrak{p}\colon F\longrightarrow\mathbb{R}_{\geq 0}$, given by $\nu_p(x):=\mathsf{card}(\mathcal{O}_F/\mathfrak{p})^{\mathsf{ord}_\mathfrak{p}(x)}$. We denote by $F_\mathfrak{p}$ the completion of $F$ with respect to the valuation $\nu_\mathfrak{p}$. I wuold prove that ring of integers of $F_\mathfrak{p}$, which we denote by $\mathcal{O}_{F,\mathfrak{p}}$ is a Dedekind domain and that the valuation $\overline{\nu}_\mathfrak{p}$ over $F_\mathfrak{p}$, which extends $\nu_\mathfrak{p}$, is given by $$\overline{\nu}_\mathfrak{p}(x)=\mathsf{card}(\mathcal{O}_{F,\mathfrak{p}}/\mathfrak{p}\mathcal{O}_{F,\mathfrak{p}})^{\mathsf{ord}_\mathfrak{p}(x)}.$$ I don't know if this result is really true, however, conviced by this at pag. $134$, $(4.1)$, I think that the result is true. If the result isn't true can anyone explain me what is $\mathsf{ord}_\mathfrak{p}(\alpha_v)$ in $(4.1)$ of the lecture notes linked above?

Answer 1

You are confusing the valuation $v_\mathfrak{p} = \text{ord}_\mathfrak{p}$ and the absolute value $|\alpha|_v= q^{-v(\alpha)}$.

In $\mathcal{O}_{F,\mathfrak{p}} = \varprojlim \mathcal{O}_F/\mathfrak{p}^n$

$\qquad$(its elements are sequences $(\alpha_1,\alpha_2,\ldots),\alpha_n \in \mathcal{O}_F/\mathfrak{p}^n$,$ \alpha_n \equiv \alpha_m \bmod \mathfrak{p}^m, m\le n$

$\qquad$ with the pointwise addition and multiplication)

Only the elements with $v_\mathfrak{p}(\alpha) \ge 1$ (ie. $\alpha_1 \equiv 0 \bmod \mathfrak{p}$) are not inversible.

Consequently for some $v_\mathfrak{p}(\alpha)=1$ then all the ideals are of the form $\alpha^n \mathcal{O}_{F,\mathfrak{p}}$ so that the unique factorization in prime/maximal ideals is obvious and $\mathcal{O}_{F,\mathfrak{p}}[\alpha^{-1}]$ is a field as well as $\mathcal{O}_{F,\mathfrak{p}}/\alpha \mathcal{O}_{F,\mathfrak{p}} \cong \mathcal{O}_F/\mathfrak{p}$.

After that you need to show $\mathcal{O}_{F,\mathfrak{p}}$ is a finite extension of $\mathbb{Q}_p$ and look at $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$ vs $\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$.