Reducibility behavior of polynomials with varying coefficients (Q1916886)

Let \(K\) be a number field, let \(O_K\) be its ring of integers and \(h\in K [Y]\) with \(\deg (h)= n>0\). Let \(i\) be an integer \(0\leq i\leq n\), \(i\) coprime to \(n\); let \(R\) (resp. \(N)\) be the set of \(a\) in \(O_K\) such that \(h(Y)- aY^i\) is reducible (resp. has a root in \(K) \). The author compares \(R\) and \(N\). In general, \(R-N\) is finite and the paper is devoted to the rare cases where this is wrong. With the help of a theorem of \textit{C. L. Siegel} [Abhandlung Akad. Berlin 1929, No. 1 (1929; JFM 56.0180.05)] and a method of \textit{M. Fried} [J. Number Theory 6, 211-231 (1974; Zbl 0299.12002)], the author reduces the problem to precise properties of some finite groups and gives a satisfying answer.