## Additive and multiplicative relations connecting conjugate algebraic numbers.(English)Zbl 0586.12001

The author investigates the possibility of solving equations of the form (A) $$\alpha_1^{a_1}\cdots\alpha_k^{a_k}=1$$ and (B) $$a_1\alpha_1+\ldots+a_k\alpha_k=0$$, where $$a_1,\ldots,a_k$$ are given non-zero integers with greatest common factor 1 and where each $$\alpha_ i$$ is to be a conjugate of some algebraic integer $$\alpha$$ (with the $$\alpha_i$$ not necessarily distinct). A solution of (A) (resp. (B)) is trivial if $$\alpha$$ is a root of unity (resp. $$\alpha =0)$$. He shows that (A) and (B) have a non-trivial solution if and only if there is an $$n\geq k$$ and an $$n\times n$$ incomplete Latin square of determinant zero, each row and column being a permutation of $$a_1,\ldots, a_k$$ and $$(n-k)$$ zeros. This result has the unexpected consequence that (A) is solvable if and only if (B) is solvable.
He gives a computationally more convenient formulation of the criterion in terms of sets of rational integer solutions to $$a_1x_1+\ldots+a_kx_k=0$$ and applies this result to a number of interesting examples.

### MSC:

 11R04 Algebraic numbers; rings of algebraic integers 11D57 Multiplicative and norm form equations 05B15 Orthogonal arrays, Latin squares, Room squares
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### References:

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