Totally p-Adic Numbers of Small Height in an Abelian Extension of ${{\mathbb {Q}}}$

Abstract

Let p be prime, and ${{\mathbb {Q}}}_p$ the field of p-adic numbers. We say that a number is totally p-adic if its minimal polynomial splits completely over ${{\mathbb {Q}}}_p$. For a particular prime p and degree d, what can we say about the smallest nonzero height of an algebraic number of degree d that splits completely over ${{\mathbb {Q}}}_p$? If $d=2$, we have that the smallest nonzero height is either $\tfrac{1}{2} \log \left( \tfrac{1+\sqrt{5}}{2}\right)$ if $p \equiv 1, 4 \pmod 5$, and $\tfrac{1}{2} \log 2$ otherwise. We prove, for any degree d, a congruence condition exists if we restrict to algebraic numbers that exist in an abelian extension of ${{\mathbb {Q}}}$.