Bogomolov property for certain infinite non-Galois extensions

For an algebraic number, let h () 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 Q lie in a finite local extension K_/Qₚ, then we obtain a lower bound for h (). We also discuss the lower bound for canonical height of an elliptic curve E/Q in such infinite extensions.