Talk:P-adic Norm not Complete on Rational Numbers

This proof begins by requiring an integer $a$ where $1 \le a \lt p$, and then later requires $a \ne 1$ and $a \ne p - 1$, so $1 \lt a \lt p - 1$, which precludes $p$ being $2$ or $3$. The constraints on $a$ seem to be necessary for the proof, so the proof doesn't seem to hold for $p = 2$ or $3$. --Leigh.Samphier (talk) 03:34, 19 August 2018 (EDT)

If you can see your way towards fixing this up, feel free. --prime mover (talk) 06:22, 19 August 2018 (EDT)

$\left\vert{\,\cdot\,}\right\vert_p$ versus $\left\Vert{\,\cdot\,}\right\Vert_p$

Re: the issue of $\left\vert{\,\cdot\,}\right\vert_p$ versus $\left\Vert{\,\cdot\,}\right\Vert_p$, I've found that the most of the popular texts on p-adic numbers all use the notation $\left\vert{\,\cdot\,}\right\vert_p$, as do most on-line references. I presume that this is to remind the reader of the connection of the p-adic absolute value with the the usual absolute value on the Reals, and to avoid possible confusion with the p-norm on the p-sequence space. Since Proofwiki has used the notation $\left\Vert{\,\cdot\,}\right\Vert_p$ for the p-adic norm, I'm happy to change this theorem to conform to this. Just let me know. --Leigh.Samphier (talk) 09:34, 11 September 2018 (EDT)

Please use whatever is standard on $\mathsf{Pr} \infty \mathsf{fWiki}$. As has been pointed out on another thread, everything on this appalling site is completely wrong, so there's absolutely no point even bothering to try to get anything right. --prime mover (talk) 09:59, 11 September 2018 (EDT)

I read that as 'Please use whatever is standard on ProofWiki.' So I will change the use of $\left\vert{\,\cdot\,}\right\vert_p$ to $\left\Vert{\,\cdot\,}\right\Vert_p$. Do you agree? --Leigh.Samphier (talk) 17:13, 11 September 2018 (EDT)

I don't know because I haven't worked on this area. --prime mover (talk) 17:36, 11 September 2018 (EDT)

Yes, I agree. Also, in my experience there is not a lot of information on $p$-adic norms on PW at the moment, so if you feel up for covering a source that deals with them, that'd be great! — Lord_Farin (talk) 14:56, 12 September 2018 (EDT)