Why is every nonarchimedean local field of characteristic zero a finite extension of $\mathbb{Q}_p$?

Question

Let $K$ be a field of characteristic zero that is complete with respect to the multiplicative valuation $|\cdot|:K \to \mathbb{R}$. Also assume that this multiplicative valuation is nonarchimedean, that is, $$ |x+y| \le \max\{|x|,|y|\} $$ This valuation induces a metric topology on $K$, and we also assume that $K$ is locally compact. I understand the following: $K$ must contain (an isomorphic copy of) $\mathbb{Q}$, and the restriction of the valuation to $\mathbb{Q}$ must be $|\cdot|_p$ for some $p$ (by Ostrowski's theorem), and the closure of $\mathbb{Q}$ in $K$ must be $\mathbb{Q}_p$.

The only assertion left is to say that $[K:\mathbb{Q}_p] < \infty$. According to Milne, "If $K$ has infinite degree over $\mathbb{Q}_p$, it will not be localy compact." Can someone explain why this is true?

Answer 1

A comment above suggests using Krasner's lemma to explain the finite-dimensionality. I will describe a more constructive argument, which is similar to what can be found in Gouvea's book on $p$-adic numbers when he works out $\pi$-adic expansions of finite extensions of $\mathbf Q_p$, but here I will be starting from a local compactness assumption instead of a finite-dimensionality assumption.

1. Local compactness of $K$ implies compactness of its ring of integers $\mathcal O_K =\{x \in K : |x| < 1\}$ by a scaling argument (here we rely on nontriviality of the absolute value on $K$, a consequence of its local compactness and it being infinite). Therefore every subsequence of $\mathcal O_K$ has a convergent subsequence.

2. ("discreteness of absolute value") There is a nonzero $\pi \in K$ making $|\pi|$ the biggest absolute value less than 1: $|\pi| = \max\{|x| : |x| < 1\}$. Indeed, if there are no such maximum then there is an increasing sequence of absolute values $|x_1| < |x_2| < \ldots < 1$ and the sequence $\{x_i\}$ has no convergent subsequence (a sequence with a nonzero limit must have eventually constant absolute value but all the $x_i$'s have distinct absolute values).

3. ("ramification index") By item 2, $|K^\times| = |\pi|^\mathbf Z$. Therefore $|p| = |\pi|^e$ for some $e \in \mathbf Z$. Since $|p| < 1$ and $|\pi| < 1$, $e \in \mathbf Z^+$. Then $[|K^\times|:|\mathbf Q_p^\times|] = [|\pi|^{\mathbf Z}:|\pi|^{e\mathbf Z}] = e$, so we have an interpretation of $e$ as an index of value groups.

4. ("residue field degree") The residue field $\mathcal O_K/\mathfrak m_K = \mathcal O_K/\pi\mathcal O_K$ has to be finite, since representatives for the residue field in $\mathcal O_K$ are all at distance 1 from each other and therefore wouldn't have a convergent subsequence if it were infinite. This residue field contains $\mathbf F_p$, so we have a finite residue field degree $f = \dim_{\mathbf F_p}(\mathcal O_K/\mathfrak m_K)$.

5. ("some $\pi$-adic expansion") Let $S$ be a set of representatives for the residue field, with $0$ representing the zero element of the residue field, so $S$ contains $p^f$ elements. In the same way $p$-adic expansions in $\mathbf Z_p$ can be developed from the representatives $\{0,\ldots,p-1\}$ of the residue field of $\mathbf Q_p$, and they are unique, every element of $\mathcal O_K$ has a unique $\pi$-adic expansion $\sum_{j \geq 0} s_j\pi^j$ with $s_j \in S$ (uniqueness uses $0 \in S$). For nonzero $x \in K$, $|x| = |\pi|^n$ for some $n \in \mathbf Z$ (possibly $n < 0$), so $|x/\pi^n| = 1$. Then writing the $\pi$-adic expansion of $x/\pi^n$ and multiplying through by $\pi^n$ afterwards, each nonzero element of $K$ has a (unique) $\pi$-adic expansion $\sum_{j \geq n} s_j\pi^j$ with $s_n \not= 0$.

6. ("good choices") Lift a basis of the residue field over $\mathbf F_p$ into $\mathcal O_K$: there is a set $\{b_1,\ldots,b_f\}$ in $\mathcal O_K$ such that they reduce modulo $\mathfrak m_K$ to an $\mathbf F_p$-basis of the residue field. We can use as $S$ the set of linear combinations $\{a_1b_1 + \cdots + a_fb_f : a_i \in \{0,\ldots,p-1\}\}$. In the $\pi$-adic expansions of step 5, we don't need to use the powers $\pi^j$ to represent all nonzero absolute values: write an integer $j$ as $eq+r$ with $q \in \mathbf Z$ and $0 \leq r \leq e-1$, so $|\pi|^j = |\pi|^{eq+r} = |\pi|^{eq}|\pi^r| = |p^q\pi^r|$. Therefore we can use the numbers $p^q\pi^r$ to represent all nonzero absolute values in $K$, with $|p^q\pi^r| \leq 1$ if and only if $q \geq 0$ and $r \geq 0$.

7. ("basis over $\mathbf Q_p$") Using step 6, each element of $\mathcal O_K$ can be written as a series $\sum_{q \geq 0} \sum_{r=0}^{e-1} s_{qr}p^{q}\pi^r$ where $s_{qr} = a_{1qr}b_1 + \cdots + a_{fqr}b_f$ for some $a_{iqr} \in \{0,\ldots,p-1\}$. Thus each element of $\mathcal O_K$ has the form $$ \sum_{q \geq 0} \sum_{r=0}^{e-1} \sum_{i=1}^f a_{iqr}b_ip^{q}\pi^r = \sum_{r=0}^{e-1} \sum_{i=1}^f\left(\sum_{q \geq 0} a_{iqr}p^{q}\right)b_i\pi^r = \sum_{r=0}^{e-1} \sum_{i=1}^fc_{ir}b_i\pi^r $$ where $c_{ir} = \sum_{q \geq 0} a_{iqr}p^{q}$ is a $p$-adic integer: $c_{ir} \in \mathbf Z_p$. For nonzero $x \in K$, $|p^Nx| < 1$ for some (large) integer $N$, so we can write $p^Nx$ in the above form and then divide by $p^N$, absorbing that $p^N$ into the coefficients $c_{ir}$ to make them lie in $\mathbf Q_p$ (possibly no longer $\mathbf Z_p$). Thus each element of $K$ can be written as a $\mathbf Q_p$-linear combination of the finite set $\{b_i\pi^r : 1 \leq i \leq f, 0 \leq r \leq e-1\}$, which proves $K$ is finite-dimensional over $\mathbf Q_p$ with dimension at most $ef$. In fact, that finite set is linearly independent over $\mathbf Q_p$, so $[K:\mathbf Q_p] = ef$, which is no surprise ("$n=ef$").