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!
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!