## Polynomials with nontrivial relations between their roots.(English)Zbl 0881.12001

Let $$K$$ be a subfield of the field $$\mathbb{C}$$ of complex numbers, $$f(X)$$ be an irreducible polynomial over $$K$$, and $$\Omega\subseteq \mathbb{C}$$ be the set of roots of $$f(X)$$. Consider the Galois group $$G:=\text{Gal} (K(\Omega)/K)$$ as a (transitive) permutation group acting on $$\Omega$$. The paper considers connections between $$G$$ and the existence of nontrivial multiplicative and additive relations on $$\Omega$$ of the form $$\prod_{a\in\Omega} \alpha m_\alpha=c$$ or $$\sum_{\alpha\in\Omega} c_\alpha \alpha=c$$ with integer $$m_\alpha$$ and $$c$$, $$c_\alpha\in K$$. Earlier results along these lines have been obtained by C. J. Smyth [J. Number Theory 23, 243-254 (1986; Zbl 0586.12001)]. Some typical results of the present paper are the following.
Theorem 1. If $$G$$ acts 2-transitively on $$\Omega$$ and there is a nontrivial multiplicative relation between the roots of $$f(X)$$, then $$p:= \deg f(X)$$ is prime and $$G$$ is isomorphic to the affine group of order $$p(p-1)$$. Moreover, if $$K$$ is real and $$p>2$$, then $$f(X)$$ has the form $$X^p-b$$ for some $$b\in K$$.
Theorem 2. Suppose $$K=\mathbb{Q}$$, $$f(X)\in \mathbb{Z}[X]$$ and $$G$$ acts primitively on $$\Omega$$. If $$\deg f(X)$$ is not prime and there is a nontrivial multiplicative relation between the roots of $$f(X)$$, then $$f(X)$$ is reducible modulo $$p$$ for each prime $$p$$ not dividing the leading coefficient of $$f(X)$$.
Theorem 5. Suppose that $$G$$ contains a transitive subgroup $$M$$ where either $$M$$ is abelian or $$M$$ has an abelian subgroup of index 2. If the roots of $$f(X)$$ are not roots of unity and there are distinct $$\alpha$$, $$\beta$$ and $$\gamma$$ in $$\Omega$$ such that $$\alpha\beta\gamma^{-1}\in K$$, then $$|M|$$ is divisible by 6 or 10 (and 6 divides $$|M|$$ if $$M$$ is abelian). This generalizes a theorem of M. Drmota and M. Skalba [Acta Arith. 71, 65-77 (1995; Zbl 0818.11038)].
Corollary to Theorem 3’. If $$K$$ is real then no root of $$f(X)$$ lies in the $$K$$-convex hull of the remaining roots.
Reviewer: J.D.Dixon (Ottawa)

### MSC:

 12E10 Special polynomials in general fields 12F12 Inverse Galois theory

### Keywords:

multiplicative relation; roots; Galois group; additive relations

### Citations:

Zbl 0586.12001; Zbl 0818.11038
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