Algebraic closure of p-adic rationals, $\overline{\mathbb Q}_p$, and its completion, $\Omega_p$, are not locally compact

Question

Trying to show $\overline{\mathbb Q}_p$ and $\Omega_p$ are not locally compact.
I can prove it by showing that the unit sphere is not locally compact.
That is to say, any sequence on the unit sphere has no converging sub-sequence.

I tried to take a sequence of distinct roots of unity but I can't seem to move pass that.
Any ideas?

Answer 1

For a field $K$ with non-archimedean valuation $v$, if $K$ is locally compact, then $K$ is complete. To prove this, note that $K$ is locally compact iff $O_K$ is compact. Since the topology on $O_K$ is defined by a metric, this gives $O_K$ is complete. So $K$ is complete. This solves $\overline{\mathbb{Q}_p}$.

For $\Omega_p$, we appeal to another consequence of locally compact: that $v$ must be a discrete valuation. Since $O_K$ is compact, the quotient $O_K/I_a$ is compact for any $a>0$, where $I_a$ the ideal $\{v>a\}$. But $O_K/I_a$ is discrete (since we live in an ultrametric space), so there are only finitely many points. This shows $v(O)-\{0\}$ has a smallest element. But of course the valuation on $\Omega_p$ is not discrete (it already fails for $\overline{\mathbb{Q}_p}$).

In fact, we have

Theorem Let $K$ be a field with non-archimedean valuation $v$. Then $K$ is locally compact iff the following conditions are satisfied:

1. $K$ is complete;
2. $v$ is a discrete valuation; and
3. the residue field $k$ is finite.