# Any continuous non-constant map from $Q_p$ to $R$

by user45765   Last Updated September 11, 2019 18:20 PM - source

Consider $$Q_p$$ and $$R$$ where $$Q_p$$ is $$p-$$adic numbers with $$p-$$adic topology. If $$Q_p\to R$$ is a ring homomorphism, then certainly it must be trivial as $$Z\to Z$$ inducing $$Q\to Q$$ but infinite series of p-adic element does not converge in $$R$$ in general.

$$\textbf{Q:}$$ It is natural to ask whether one can classify all continuous maps $$Hom_{Cts}(Q_p,R)$$ besides constant maps. It is certainly possible to get just set theoretical embedding by axiom of choice. Does $$Hom_{Cts}(Q_p,R)$$ have non-constant functions? I do not see an obvious choice of candidate for non-constant maps.

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There are many continuous, non-constant functions; for example, the $$p$$-adic norm is one such (as is any positive power of the $$p$$-adic norm).