Polynomials with roots in $\mathbb {Q}_{p}$ for all $p$

Abstract:

Let $f(x)$ be a monic polynomial in $\mathbb {Z}[x]$ with no rational roots but with roots in $\mathbb {Q}_{p}$ for all $p$, or equivalently, with roots mod $n$ for all $n$. It is known that $f(x)$ cannot be irreducible but can be a product of two or more irreducible polynomials, and that if $f(x)$ is a product of $m>1$ irreducible polynomials, then its Galois group must be a union of conjugates of $m$ proper subgroups. We prove that for any $m>1$, every finite solvable group that is a union of conjugates of $m$ proper subgroups (where all these conjugates have trivial intersection) occurs as the Galois group of such a polynomial, and that the same result (with $m=2$) holds for all Frobenius groups. It is also observed that every nonsolvable Frobenius group is realizable as the Galois group of a geometric, i.e. regular, extension of $\mathbb {Q}(t)$.