Two remarks on the inverse Galois problem for intersective polynomials

Abstract

Abstract. A (monic) polynomial f (x) ∈ Z[x] is called intersective if the congruence f (x) ≡ 0 mod m has a solution for all positive integers m. Call f (x) nontrivially intersective if it is intersective and has no rational root. It was proved by the author that every finite noncyclic solvable group G can be realized as the Galois group over Q of a nontrivially intersective polynomial (noncyclic is a necessary condition). Our first remark is the observation that the corresponding result for nonsolvable G reduces to the ordinary inverse Galois problem for G over Q. The second remark has to do with the scarcity of explicit examples of nontrivial intersective polynomials with given Galois group, and gives the first known example for the dihedral group of order ten.

The Hasse Principle for Lines on del Pezzo Surfaces

“…For non-cyclic groups however, counter-examples to the Hasse principle always occur. Namely, let k be a number field and let Γ be a non-cyclic group which is realisable as a Galois group over k. Then it was shown in [46] and [47] that there exists some finite étale scheme over k whose splitting field has Galois group Γ and which is a counter-example to the Hasse principle.…”

“…See [2], [46] and [47] for other examples. Our first result states that such schemes can indeed occur as the Hilbert scheme of lines on a cubic surface.…”

Covering a semi-direct product and intersective polynomials

“…The theory of intersective polynomials is described in detail by Sonn [17,18] or by Rabayev and Sonn [16]. Intersective polynomials have application in combinatorial number theory.…”

On intersective polynomials with non-solvable Galois group

“…In [21], Sonn shows that intersective polynomials exist for all non-solvable groups that occur as Galois groups over Q. He also shows in [20,Theorem 2] that for solvable groups that admit a k-covering of subgroups with the trivial intersection property of Prop.…”