Two remarks on the inverse Galois problem for intersective polynomials (Q1032653)

A nontrivially intersective polynomial with integral coefficients is a polynomial that has a root mod \(m\) for every integer \(m\) but does not have a rational root. It follows from a previous paper of the author [Proc. Am. Math. Soc. 136, No. 6, 1955--1960 (2008; Zbl 1195.12007)] that a solvable group appears as a Galois group of a nontrivially intersective polynomial if and only if it is noncyclic. The first remark provides an answer for the remaining groups: a nonsolvable group appears as a Galois group of a nontrivially intersective polynomial if and only if it appears as a Galois group over \(\mathbb Q\). Explicit known examples of nontrivially intersective polynomials are rare. The second remark gives such a polynomial with Galois group \(D_{10}\), the dihedral group of order 10.