Unramified extensions over low degree number fields

Abstract

For various nonsolvable groups G, we prove the existence of extensions of the rationals Q with Galois group G and inertia groups of order dividing ge(G), where ge(G) is the smallest exponent of a generating set for G. For these groups G, this gives the existence of number fields of degree ge(G) with an unramified G-extension. The existence of such extensions over Q for all finite groups would imply that, for every finite group G, there exists a quadratic number field admitting an unramified G-extension, as was recently conjectured. We also provide further evidence for the existence of such extensions for all finite groups, by proving their existence when Q is replaced with a function field k(t) where k is an ample field.On the other hand, It is well known that every finite group G appears as a Galois group of an unramified extension over some number field. This is obtained by realizing G as a Galois group of a tame (tamely ramified) extension L 0 /K over some number field, and finding a (not necessarily Galois) number field M which is disjoint from L 0 and satisfies: "for every prime P of M, the ramification index of P over its restriction p to K is divisible by the ramification index of p in L 0 ". 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

On Galois extensions with prescribed decomposition groups

“…The following theorem relaxes the assumptions of Proposition 4.1, allowing applications for quite a number of almost-simple groups. Its argument is similar to Theorem 4.4 in [13].…”

“…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.…”

Unramified extensions of quadratic number fields with certain perfect Galois groups

“…For any finite group G, there exist infinitely many quadratic number fields K such that K possesses a Galois extension with Galois group G unramified at all (finite and infinite) primes of K. This is known to be true only in some special cases, including the case of cyclic groups (from class field theory, cf. [1]), symmetric and alternating groups (e.g., [22] (which does not consider ramification at infinity) or [6] (which does)), and a few other almost simple finite groups (e.g., [10]). The common idea for the nonsolvable cases is to carefully specialize suitable function field extensions over Q (or equivalently, multiparameter polynomials over Q) with prescribed Galois group to obtain extensions of Q with a desired ramification behavior.…”

“…In order to avoid too much technicality, we only refer to [11,Theorem 2.2] for an effective version of Theorem 3.2. b) For many applications, it is enough to obtain some specialization values a ∈ P 1 (K) yielding a certain prescribed local behavior, rather than controlling the behavior for all specialization values. In this context, the notion of a universally ramified prime (see [2] or [10]) becomes useful: let U(N/K(t)) be the set of all primes of K ramifying in all specializations N a /K where a ∈ P 1 (K) is a non-branch point of N/K(t). Let S be any finite set of primes of K disjoint from U(N/K(t)).…”

On Unramified Solvable Extensions of Small Number Fields

“…We additionally define e(Q, G) as the minimal number e such that Q admits a tamely ramified G-extension all of whose ramification indices divide e. The relevance of this definition for the original question on unramified G-extensions is due to Abhyankar's lemma, which shows immediately that d(Q, G) ≤ e(Q, G) (cf. [8,Lemma 2.1]). Note that while bounds on e(Q, G) do not automatically yield bounds on d ′ (Q, G) in general, they do as soon as the implied tamely ramified G-extensions satisfy certain additional local conditions, cf.…”