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Here is a finite-ness result for $p$-adic fields. Let $k$ be a p-adic field. One knows that there are only finitely many square classes of $k$ -- that is $k^{\times} / k^{\times 2}$ is finite (as a set). Thus the map $k \rightarrow k$ given by $x \mapsto x^2$ misses only finitely many points. I think a A similar situation holds should hold for $p$-th classes p^{th}$-classes of$l$-adic fields. As a concrete example, we know that for$k = \mathbb{Q}_2$, the square classes$k^\times/k^{\times 2}$are generated by$2,-1,5$(I think), 2,-1,5$, in particular, $k^{\times} / k^{\times 2} \cong (\mathbb{Z}/2)^{\times 3}$.