Multiplicative relations with conjugate algebraic numbers

Abstract

We study what algebraic numbers can be represented by a product of algebraic numbers conjugate over a fixed number field K in fixed integer powers. The problem is nontrivial if the sum of these integer powers is equal to zero. The norm of such a number over K must be a root of unity. We show that there are infinitely many algebraic numbers whose norm over K is a root of unity and which cannot be represented by such a product. Conversely, every algebraic number can be expressed by every sufficiently long product in algebraic numbers conjugate over K. We also construct nonsymmetric algebraic numbers, i.e., algebraic numbers such that no elements of the corresponding Galois group acting on the full set of their conjugates form a Latin square.