# 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.

Published

25.07.2007

How to Cite

*Ukrains’kyi Matematychnyi Zhurnal*, Vol. 59, no. 7, July 2007, pp. 890–900, https://umj.imath.kiev.ua/index.php/umj/article/view/3354.

Issue

Section

Research articles