Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 28 / 1 / b / ii / Solution

The compactness and completeness proved above apply again. The finite ring $\mathcal{O}/t\mathcal{O}$ has a positive integer characteristic $m$, so $m\in t\mathcal{O}$ and $|m|<1$. Prime factorization supplies a prime $p$ with $|p|<1$. In characteristic zero the restriction to $\mathbb{Q}$ is therefore a positive power of the p-adic absolute value, by the preceding classification. Completing $\mathbb{Q}$ embeds the P-adic number field ${\mathbb{Q}}_p$ into the complete field $K$.

Now $p\mathcal{O}$ is open, so $\mathcal{O}/p\mathcal{O}$ is finite. It is an ${\mathbb{F}}_p$-vector space; choose lifts $e_1,\dots ,e_d\in \mathcal{O}$ of a basis. Successively reducing modulo $p\mathcal{O}$ gives, for every $x\in \mathcal{O}$,Convergence in $K$ follows from $|p|<1$. Every element of $K$ becomes integral after multiplication by a power of $p$, so $K={\sum }_j{\mathbb{Q}}_p{e}_j$. Consequently $\boxed{[K:{\mathbb{Q}}_p]<\infty }$: $K$ is a p-adic field. This finite-module proof does not assume discreteness of the original value group in advance.