How to show that $\mathbb{Q}_p^*$ is totally disconnected?

Question

Let $\mathbb{Q}_p$ be the field of p-adic numbers and $\mathbb{Q}_p^*$ the set of invertible elements in $\mathbb{Q}_p$.

How to show that $\mathbb{Q}_p^*$ is totally disconnected? Thank you very much.

Answer 1

A subspace of a totally disconnected space is disconnected, so you need to worry only about $\mathbb{Q}_p$. Try showing that the connected component $C$ of some arbitrary element $q \in \mathbb{Q}_p$ is $\displaystyle \{ q \}$. This can be done by obtaining a contradiction if you assume $C$ has only one point. Try writing it as a union of two disjoint open sets.

Remark: Just realized I have given a very general strategy, not much to do with $\mathbb{Q}_p$. It helps in the proof if you know that the open balls in $\mathbb{Q}_p$ are both open and closed.

Just wanted to try the spoiler technique, sorry....

If $C$ consists of a point other than $q$, then there exists an integer $z$ such that $B(q,p^z) \cap C \neq C$, so then we have $ C = [(B(q,p^z) \cap C] \cup [(\mathbb{Q}_p\setminus (B(q,p^z))\cap C] $, a contradiction.