Why $ev_p$ is normalized, where $v_p$ is the $p$-adic valuation of $\mathbb{Q}_p$ being extended to $K$? ( Neukirch's ANT book, (5.5) Proposition)

Question

I am reading the Neukirch's Algebraic number theory, p.138, proof of the II-(5.5) proposition and stuck at some point.

(5.5) Proposition. Let $K|\mathbb{Q}_p$ be a $\mathfrak{p}$-adic number field with valulation ring $\mathcal{O}$ and maximal ideal $\mathfrak{p}$, and let $p\mathcal{O}=\mathfrak{p}^e$. Then the power series $$ \exp(x)=1+x+\frac{x^2}{2 !}+\frac{x^3}{3!}+\cdots \ \operatorname{and} \ \log(1+z)=z-\frac{z^2}{2}+\frac{z^3}{3}-\cdots,$$ yield, for $n> \frac{e}{p-1}$, two mutually inverse isomorphisms ( and homeomorphisms ) $$ \mathfrak{p}^n \xrightarrow{\exp} U^{(n)}, \ U^{(n)} \xrightarrow{\log} \mathfrak{p}^n.$$

Proof of (5.5) : We again think of the $p$-adic valuation $v_p$ of $\mathbb{Q}_p$ as being extended to $K$. Then $v_{\mathfrak{p}} =ev_p$ is the normalized valuation of $K$. ( Next proof is omitted..)

Why the statement is true? Here $v_{\mathfrak{p}}$ seems to the normalized exponential valuation of $K$. How the ramification index $e$ is involved? In fact, in the linked question that I proposed, Understanding the Neukirch, Algebraic Number Theory, p.142, (5.8) Corollary., 'SomecallmeTim' showed that $v_{\mathfrak{p}}(\alpha) =ev_p(\alpha)$ for all $\alpha \in \mathbb{Z}_p$ and I don't know how to generalize this equality for all $\alpha \in K$.

EDIT : I think that I generalize this equality for all $\alpha \in K$ : see edit ( I edited ) of 'SomecallmeTim' 's answer in the linked question.

My other strategy is as follows.

By line 2 in the Neukrich's book p.135, proof of (5.2) Proposition, $K$ is endowed with the extended valuation $ | \alpha| = \sqrt{|N_{K|\mathbb{Q}_p}(\alpha)|}^{1/d}$, where $d : = [K : \mathbb{Q}_p] $. In other words, by p.133 below statement, $$ w(\alpha) = \frac{1}{d}v_p(N_{K|\mathbb{Q}_p}(\alpha)).$$

And we show that $ew$ is normalizaed valuation of $K$. By the fundamental identity, $ d= ef $ , where $f := [\mathcal{O}_K/\mathfrak{p} : \mathbb{Z}_p/p\mathbb{Z}_p ] = [ \mathcal{O}_K /\mathfrak{p} : \mathbb{Z}/p\mathbb{Z}]=[\mathcal{O}_K / \mathfrak{p} : \mathbb{F}_p]$ is the inertia degree. So $$ ew(\alpha) = \frac{1}{f}v_p(N_{K|\mathbb{Q}_p}(\alpha)).$$

Now, my question is,

$$ v_p(N_{K|\mathbb{Q}_p}(K^{*})) = f\mathbb{Z} $$

? If so, then as the Nuekirch's book p.121 remark, by dividing by $f$, $ \frac{1}{f}v_p(N_{K|\mathbb{Q}_p}(\alpha))$ is normalized valuation of $K$ so that $ew$ is normalized. And the question is true? If so, how to show..?

Or is there any other route to show that $v_{\mathfrak{p}} =ev_p$? I don't understand this statement at all

Can anyone help?

Answer 1

Let's do this more generally. Let $R$ be a DVR with normalized valuation $v$ and let $L/K$ be a finite extension of the fraction field $K$ of $R$ and let $w$ be an extension of $v$ to $L$ with valuation ring $S$. Let $e=e(w/v)$ be the ramification index. Choose uniformisers $\varpi \in L$ and $\pi \in K$ for both valuations.

We want to show that $ew$ is normalized, i.e. $e\cdot w(L^\times)=\Bbb Z \Leftrightarrow w(L^\times)=\frac{1}{e}\Bbb Z$. By unique factorization, we may write any element of $L$ as $u\varpi^n$ for $n \in \Bbb Z$ and $u \in S^\times$. Applying $w$ to that, we get that $w(u\varpi^n)=nw(\varpi)$. From this, one sees that what we want to prove follows if we can prove $w(\varpi)=\frac{1}{e}$. Now consider the factorization of $\pi$ in $S$. We have, by definition of $e$: $\pi=u\varpi^e$ for some $u \in S^\times$. Now apply $w$ to that and use that $w$ is an extension of $v$, which is normalized. We get

$$1=v(\pi)=w(\pi)=w(u\varpi^e)=ew(\varpi) \Rightarrow w(\varpi)=\frac{1}{e}$$.

(Remark: In fact, this observation provides us with an alternative definition of ramification indices that is not based on uniformisers and just uses value groups: If $(L,w)/(K,v)$ is an extension of valued fields, then the ramification index is the index of the value groups: $e(w/v)=[w(L^\times):v(K^\times)]$. The advantage of this definition is that value groups always exists for valued fields (even non-discretely valued ones), while uniformisers might not.)