Bogomolov property for certain infinite non-Galois extensions

For an algebraic number α, let h(α) denote its logarithmic Weil height. In 2002, Bombieri and Zannier obtained lower bounds for Weil height for totally p-adic extensions of Q, i.e., infinite Galois extension over Q with finite local degree over a rational prime p. In this paper, we formulate and prove the non-Galois analogue of this result. In particular, if a positive proportion of conjugates of ᾱ lie in a finite local extension Kν/Qₚ, then we obtain a lower bound for h(α). We also discuss the lower bound for canonical height of an elliptic curve E/Q in such infinite extensions.