Let $$ E=\{x\in\mathbb Q_p\mid |x|_p\le p^{-2}\}\setminus\left\{p^n\mid n\in\mathbb N,n\ge2\right\} $$ and $g$ be an analytic element (in Krassner's sense) of $$ F=\{x\in\mathbb Q_p\mid |x|_p<1\}\setminus\left\{p^n\mid n\in\mathbb N\right\}. $$ Suppose that for all $x\in E$, $g(x)=0$. Can one assert that $g(x)=0$ for all $x\in F$? Thanks in advance for any answer.
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- $\begingroup$ I'm a bit confused - don't $1/p^n$ have $p$-adic norms $p^n$? Are the sets supposed to be defined some other way? $\endgroup$Ben Johnsrude– Ben Johnsrude2023-06-14 18:01:54 +00:00Commented Jun 14, 2023 at 18:01
- $\begingroup$ Yes, it was a bad copy paste. Sorry for the trouble. I modified the post. $\endgroup$joaopa– joaopa2023-06-14 18:07:20 +00:00Commented Jun 14, 2023 at 18:07
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