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)
.