# Compact subgroups of a p-adic field

Definition: A p-adic field is a finite extension of $$Q_p$$.

Question: Let $$E$$ be a p-adic field, $$G$$ is a nontrivial additive compact subgroup of $$E$$, how to prove: E is isomorphic to $$Z_p^n$$ for some positive integer $$n$$. This isomorphism is not only a topological group isomorphism but also a $$Z_p$$ module isomorphism.

I guess we can prove it using non-archimedean analysis, but I still don't know how to prove it.

$$G$$ is compact, thus closed in $$E$$. It is stable by multiplication by an element of $$\mathbb{Z}$$ (dense in $$\mathbb{Z}_p$$), thus is a $$\mathbb{Z}_p$$-submodule of $$E$$.

Note that there exists a finite $$\mathbb{Q}_p$$-base of the vector subspace $$V$$ spanned by $$G$$, with vectors $$a_1, \ldots, a_n$$.

Now, for each $$v \in V$$, denote $$v_i \in \mathbb{Q}_p$$ to be the coordinate of $$v$$ in the direction $$a_i$$.

Then $$v \longmapsto v_i$$ is a linear form, thus is continuous, hence $$G_i=\{g_i,\,g \in G\} \subset \mathbb{Q}_p$$ is compact. Therefore, there is a $$N>0$$ such that for each $$i$$, $$p^NG_i \subset \mathbb{Z}_p$$.

Now, let $$G’=\bigoplus_{i=1}^n{\frac{a_i}{p^N}\mathbb{Z}_p}.$$

$$G’$$ is a finitely generated $$\mathbb{Z}_p$$-module, thus is Noetherian, and since $$G$$ is a submodule of $$G’$$, $$G$$ is finitely generated over $$\mathbb{Z}_p$$.

Since $$\mathbb{Z}_p$$ is principal and $$G$$ has no torsion, and all the nontrivial quotients of $$\mathbb{Z}_p$$ are finite, $$G$$ is a power of $$\mathbb{Z}_p$$.

• Thanks for your perfect answer. But it seems to me that we don't need the fact that all the nontrivial quotients of $Z_p$ are finite, because any finitely generated torsion-free module over a PID is a free module of finite rank. This is corollary 2.4 in Conrad's paper. – Sssss Jul 8 at 12:29
• You are right, that part is indeed pointless. I was (wrongly) concerned with the idea that some quotients of $\mathbb{Z}_p$ would still be torsion-free, but that’s obviously impossible. – Mindlack Jul 8 at 12:34