Polynomials with roots in ℚ_{𝕡} for all 𝕡

Let f ( x ) f(x) be a monic polynomial in Z [ x ] \mathbb {Z}[x] with no rational roots but with roots in Q p \mathbb {Q}_{p} for all p p , or equivalently, with roots mod n n for all n n . It is known that f ( x ) f(x) cannot be irreducible but can be a product of two or more irreducible polynomials, and that if f ( x ) f(x) is a product of m > 1 m>1 irreducible polynomials, then its Galois group must be a union of conjugates of m m proper subgroups. We prove that for any m > 1 m>1 , every finite solvable group that is a union of conjugates of m 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 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 Q ( t ) \mathbb {Q}(t) .