Mathematics

The height of an algebraic number a is a measure of how arithmetically complicated a is. We say a is totally p-adic if the minimal polynomial of a splits completely over the field of p-adic numbers. In this dissertation, we investigate what can be said about the smallest nonzero height of a degree d totally p-adic number. In particular, we look at the cases that a is either contained within an abelian extension of the rational numbers, or has degree 2 or degree 3 over the rational numbers. Additionally, we determine an upper bound on the smallest limit point of the height of totally p-adic numbers for each fixed prime p.