Thesis topics on p-adic number theory for an undergraduate

Question

I've been reading and learning about p-adic numbers the past few months for my undergraduate thesis, and they interest me very much. Any suggestions on a thesis topic I could work with the p-adic numbers?

Thank you!

Answer 1

I'd consider a topic in algebraic coding: ''Modular and p-adic Cyclic Codes'', A. Robert Calderbank, N. J. A. Sloane, published 1995 in Designs, Codes, Cryptography DOI:10.1007/BF01390768

Answer 2

A fascinating topic is Quadratic forms and the Hasse-Minkowski Theorem. Here we see how to solve certain quadratic equations in several variables over the rational numbers by considering local solutions over all $p$-adic fields, including the real number, for $p=\infty$. Have a look, for example, at this undergraduate project. You could generalise it to finite extensions of $\Bbb{Q}$.