2017/10/23 by Grizzard, Robert, Vaaler, Jeffrey D.
#11G50 #11J25 #11R04 #46B04 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1710.08399
Let K/ℚ be an algebraic extension of fields, and let α\not= 0 be contained in an algebraic closure of K. If α can be approximated by roots of numbers in K× with respect to the Weil height, we prove that some nonzero integer power of α must belong to K×. More generally, let K1, K2, … , KN, be algebraic extensions of \mathbQ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If α\not= 0 can be approximated by a product of roots of numbers from each Kn with respect to the Weil height, we prove that some nonzero integer power of α must belong to the multiplicative group K1× K2× ⋯ KN×. Our proof of the more general result uses methods from functional analysis.