# 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.

**English version** (Springer):
Ukrainian Mathematical Journal **59** (2007), no. 7, pp 984-995.

**Citation Example:** *Dubickas A.* Multiplicative relations with conjugate algebraic numbers // Ukr. Mat. Zh. - 2007. - **59**, № 7. - pp. 890–900.

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