Let p be prime, and denote by Qp the field of p-adic numbers. We say that a number is totally p-adic if its minimal polynomial splits completely over Qp. It is certain that for any particular prime p and degree d, there is a smallest non trivial height of algebraic numbers that are totally p-adic and of degree d. When do we know what that smallest height is? When d = 2, this can be answered by reducing p modulo 5. But what about when d = 3? Is there a similar congruence condition then? What happens if we restrict to only considering algebraic numbers that are contained within abelian extensions of Q? This talk will outline the various solution methods in each of these cases.
Totally p-adic Numbers of Small Height
Emerald Tatiana Stacy, Washington CollegeAuthors: Emerald Tatiana Stacy
2023 AWM Research Symposium
Number Theory at Primarily Undergraduate Institutions [Organized by Bella Tobin and Leah Sturman]