On intersective polynomials with non-solvable Galois group

Abstract

We present new theoretical results on the existence of intersective polynomials (that is, integer polynomials with roots in all Q p , but not in Q) with certain prescribed Galois groups, namely the projective and affine linear groups P GL 2 (ℓ) and AGL 2 (ℓ) as well as the affine symplectic groups AGSp 4 (ℓ) := (F ℓ ) 4 ⋊ GSp 4 (ℓ). For further families of affine groups, existence results are proven conditional on the existence on certain tamely ramified Galois extensions. We also compute explicit families of intersective polynomials for certain non-solvable groups.

Galois realizations with inertia groups of order two

“…In this section we show how Theorem 2.2 yields the existence of infinitely many optimally intersective realizations of A 5 and P SL 3 (3) over Q. This is already known for P SL 2 (7) [5].…”

“…Accordingly, a Galois extension K/Q with Galois group G is called an optimally intersective realization of G if it is the splitting field of a polynomial which is optimally intersective for G. We apply the order two inertia result above to A 5 and P SL 3 (3) to prove the existence of infinitely many optimally intersective Galois realizations of A 5 and P SL 3 (3) over Q. This has already been done by the first author for P SL 2 (7) [5]. Such results for D 5 and A 4 have appeared earlier [8], [9].…”

“…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]).…”

On Galois extensions with prescribed decomposition groups

“…Every tamely ramified extension of Q in which all ramification indices are ≤ 2 is automatically locally abelian, since the decomposition groups at ramified primes are then (order 2) central extensions of cyclic groups, i.e., abelian. Using results from the literature about Galois realizations with inertia groups of order 2, this shows for example that S n , A 5 , P SL 2 (7), P SL 2 (11), M 11 and several more almost simple groups are quotients of Gal(Q loc−ab /Q). Compare [13] for results on such Galois realizations.…”

“…with Galois group G which possess a root in every completion of K, but not in K. It is easy to see that intersective polynomials exist for every group which occurs as a Galois group (apart from cyclic groups of prime power order, for which there is a trivial group-theoretical obstruction); however, there is a group-theoretical lower bound for the number of irreducible factors of such a polynomial (the covering number of the group G), and it is in general an open problem whether this lower bound can be obtained. However, from elementary considerations, a G-realization in which all decomposition groups are cyclic, immediately yields a positive answer for the latter question (see [19] for these and related considerations, and [11] for some constructions of families of minimally intersective polynomials with non-solvable Galois groups).…”