Given a degree d and a rational prime p, we search for the smallest non-trivial height of a totally p-adic number of degree d. If d = 2, then the smallest height depends on the congruence of p mod 5. If d = 3, then instead of a congruence condition on p, we have an algorithm which can find the smallest height for a particular prime. What is so different about degree 2 numbers and degree 3 numbers? All degree 2 numbers have abelian Galois group, whereas the same cannot be said for algebraic numbers of degree 3. It can be shown that given a degree d, that the smallest non-trivial height of a totally p-adic number with abelian Galois group can be determined by a congruence condition on p.