2018

Том 70

№ 12

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# Dubickas A.

Articles: 1

Article (English)

Dubickas A.

Ukr. Mat. Zh. - 2007. - 59, № 7. - pp. 890–900

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.