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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
Full Text: DOI EuDML