 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.