On Galois realizations of the 2-coverable symmetric and alternating groups

TL;DR

This paper constructs explicit polynomials over the rationals whose Galois groups are 2-coverable symmetric and alternating groups, advancing the inverse Galois problem for these groups using computational tools.

Contribution

It provides explicit polynomial constructions for 2-coverable symmetric and alternating groups, solving a specific case of the inverse Galois problem with computational methods.

Findings

Explicit polynomials for symmetric groups S_n, 3<n<7.

Explicit polynomials for alternating groups A_n, 4<n<9.

Verification of Galois groups using MAGMA, PARI, and GAP.

Abstract

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) in 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 2<n<7, and the alternating group A_n is 2-coverable if and only if 3<n<9. 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.