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I am going for a research internship this summer in $p$-adic numbers. I am currently taking a number theory course, and have the essentials of a first year mathematics student, like multivariable calculus, linear algebra, probability, and a proofs course. I was curious to know if there are any problems in $p$-adic numbers that are accessible to an undergraduate like me.

Note that I am very motivated, and am willing to self-study abstract algebra and analysis as required for such problems (I have already begun with Fraleigh).
Thank you!

Marco Ripà
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Mahdi El Zein
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    A quick search on this site leads to https://mathoverflow.net/q/81342/27465, https://math.stackexchange.com/q/2914255/96384, and https://math.stackexchange.com/q/4538074/96384. – Torsten Schoeneberg Jan 19 '24 at 01:54
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    Best to ask a prof at your university. – Gerry Myerson Jan 19 '24 at 09:40
  • Isn't this internship going to present you with some open problems in $p$-adic analysis? I'm unsure what the motivation is here. – KCd Jan 19 '24 at 16:10
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    I think this oldie is a good match. I want others to comment on using that as a duplicate target also, because A) my vote would be immediately binding, B) Torsten's suggestion should/could also be used, C) I happened to post an answer on the earlier incarnation, and I don't think it's kosher to vote to close this one in such a case. – Jyrki Lahtonen Jan 19 '24 at 18:57
  • @KCd not exactly, my supervisor told me the field (p-adics) and he wants me to experiment with then numbers themselves rather than giving me a straightforward problem. I wanted to know what kind of questions there are using this post. – Mahdi El Zein Jan 19 '24 at 23:07
  • Thank you everyone for your replies – Mahdi El Zein Jan 19 '24 at 23:08

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Although I am self-taught in number theory, in recent years I proved a result that is connected with the following exercise on the commutative ring of $g$-adic integers (assume $g \in \mathbb{N}-\{0,1,2\}$.

For some given $g$ as above, find the minimum value of the positive integer $k$, in $\mathbb{Z}_g$, such that the fundamental equation $y^k=y$ includes all the solutions of $y^{k+1}=y$ (e.g., if $g=10$, then $k=5$ and we have a total of $15$ solutions - see 15 Solutions, p. 451).

You could start with a few, small, integers $g$ and look at what happens when $g$ is a prime or not. If yes, you can try to connect what you discover to the $p$-adic valuation definition (remembering that, in general, a $p$-adic number lies in $\mathbb{Z}_p$ iff its $p$-adic norm is $\leq 1$).

Have fun!

Marco Ripà
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