Let f (x) be a monic polynomial in Z [x] with no rational roots but with roots in 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 "m-coverable", i.e. a union of conjugates of m proper subgroups, whose total intersection is trivial. We are thus led to a variant of the inverse Galois problem: given an m-coverable finite group G, find a Galois realization of G over the rationals Q by a polynomial f (x) ∈ Z[x] which is a product of m nonlinear irreducible factors (in Q [x]) such that f (x) has a root in Q p for all p. The minimal value m = 2 is of special interest. It is known that the symmetric group S n is 2-coverable if and only if 3 ≤ n ≤ 6, and the alternating group A n is 2-coverable if and only if 4 ≤ n ≤ 8. In this paper we solve the above variant of the inverse Galois problem for the 2-coverable symmetric and alternating groups, and exhibit an explicit polynomial for each group, with the help of the software packages MAGMA, PARI and GAP.

There are several variants of the inverse Galois problem which involve restrictions on ramification. In this paper we give sufficient conditions that a given finite group G occurs infinitely often as a Galois group over the rationals Q with all nontrivial inertia groups of order 2. Notably any such realization of G can be translated up to a quadratic field over which the corresponding realization of G is unramified. The sufficient conditions are imposed on a parametric polynomial with Galois group G-if such a polynomial is available-and the infinitely many realizations come from infinitely many specializations of the parameter in the polynomial. This will be applied to the three finite simple groups A 5 , P SL 2 (7) and P SL 3 (3). Finally, the applications to A 5 and P SL 3 (3) are used to prove the existence of infinitely many optimally intersective realizations of these groups over the rational numbers (proved for P SL 2 (7) by the first author in [5]).

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