Multiplicative approximation by the Weil height

TL;DR

This paper establishes that algebraic numbers approximable by roots in certain fields must have some power in those fields, extending to products across multiple fields, using functional analysis techniques.

Contribution

It proves new multiplicative approximation results involving the Weil height for algebraic numbers and their products across multiple algebraic extensions.

Findings

Some nonzero power of an approximable algebraic number lies in the field.

Approximation by products of roots implies algebraic number's power belongs to a specific multiplicative group.

Uses functional analysis methods to prove the results.

Abstract

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.