Galois realizations with inertia groups of order two

Abstract

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]).

On number fields with power-free discriminant

“…a = 4 and a = 6 yield distinct squares of primes as discriminants). Finally, the same source [14] also provides suitable extensions for the group P SL 3 (2) (see Theorem 4.3), which also has been noted in [12]. 6.…”

“…Since P SL 2 (5) ∼ = A 5 , it suffices to treat the same problem for A 5 . Such extensions are contained, e.g., in [12]. For D 5 -extensions with all inertia groups of order ≤ 2, one may use a Q-regular extension given in [13].…”

Unramified extensions over low degree number fields

“…Specializing t → 1 and t → 2, gives residue extensions with coprime discriminant. For G = A 5 , a regular Gextension with ramification type (2A, 2A, 2A, 3A) and without universally ramified primes is given (via a polynomial f (0, v, x)) in [18,Theorem 3.1].…”

“…In the case G = P SL 2 (7), an extension of Q(t) with all inertia groups of order 2 and without universally ramified primes is deduced from [18, Proof of Theorem 3.2] by specializing some of the parameters. Finally, a P SL 3 (3)-extension of Q(t) with all inertia groups of order 2 is given in [18,Lemma 3.4]. Specializing that polynomial at t → 1 and t → 3 gives extensions with coprime discriminant.…”

“…Abhyankar's lemma then implies that L := L 0 M is an unramified extension of M with Gal(L/M) ∼ = G. Moreover, the resulting extension L/M is tamely defined over K, that is, there is a tame Galois extension L 0 /K such that L 0 ⊗ K M ∼ = L. This is a common method for generating unramified extensions, e.g. used in [15], [18], [9], [14], [24]. In fact, since the Inverse Galois Problem is known for many groups G, we restrict our consideration to such groups and hence choose K = Q.…”