= Solution
The restriction to $\mathbb Q$ cannot be trivial: otherwise the infinitely many integers would be pairwise distance one inside a bounded ball, contradicting local compactness. By part (c), it induces the $p$-adic topology for one prime $p$. Since a locally compact valued field is complete, the embedding $\mathbb Q\hookrightarrow K$ extends to a closed embedding $\mathbb Q_p\hookrightarrow K$.
Now $K$ is a locally compact topological vector space over the nondiscrete <local field> $\mathbb Q_p$. Such a vector space is finite-dimensional: a compact neighbourhood, together with a maximal linearly independent subset chosen at a fixed separation scale, is totally bounded only if that subset is finite, and its span is then open and closed; maximality makes it all of $K$. Thus $[K:\mathbb Q_p]<\infty$.
Solved by gpt-5.6-sol high.
Back to article page
