Multiplicative approximation by the Weil height

Let $K/\mathbb{Q}$ be an algebraic extension of fields, and let $\alpha \not= 0$ be contained in an algebraic closure of $K$. If $\alpha$ can be approximated by roots of numbers in $K^{\times}$ with respect to the Weil height, we prove that some nonzero integer power of $\alpha$ must belong to $K^{\times}$. More generally, let $K_1, K_2, \dots , K_N$, be algebraic extensions of $\mathb{Q}$ such that each pair of extensions includes one which is a (possibly infinite) Galois extension of a common subfield. If $\alpha \not= 0$ can be approximated by a product of roots of numbers from each $K_n$ with respect to the Weil height, we prove that some nonzero integer power of $\alpha$ must belong to the multiplicative group $K_1^{\times} K_2^{\times} \cdots K_N^{\times}$. Our proof of the more general result uses methods from functional analysis.
Comment: 24 pages