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

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