On Galois extensions with prescribed decomposition groups

Unramified extensions of quadratic number fields with certain perfect Galois groups

“…We remark that for the cases treated in Theorem 1.1, there are explicit polynomials available, which make a concrete enumeration of infinitely many quadratic number fields with the above property possible. In Theorem 5.1, we give such an explicit family in the case G = SL 2 (7). At the same time, it should be noted that the proof of Theorem 1.1 itself does not rely on the concrete shape of these polynomials, instead requiring only general theoretical results such as the rigidity method, and results on the behavior of inertia and decomposition groups under specialization of function field extensions (as well as the subgroup structure of the groups in question).…”

“…The extensions of Q with Galois group P GL 2 (p) constructed in the above proof are in particular locally cyclic, i.e., all decomposition groups at primes of Q are cyclic. This property was previously studied in [7], and locally cyclic extensions were obtained, e.g., for the special case G = P GL 2 (7) ([7, Theorem 5.5]). Our above result (for infinitely many primes p) in particular improves on [7, Theorem 2.1].…”

“…We specialize u → u 0 essentially as above, except that we will choose u 0 7-adically close to u = ∞ rather than to u = 0. Note that the normalizer of a subgroup of order 3 in P GL 2 (7)…”

On Unramified Solvable Extensions of Small Number Fields

“…It is also well behaved under taking wreath products, at least under some technical assumptions. J. König [7] Proof. Let ((N i , G i ) | i ∈ {1, .…”

“…Finally, we include a number-theoretic lemma which ensures that we have d ′ (Q, G) ≤ e(Q, G) under certain extra conditions. LEMMA 2.7 [7,Lemma 4.5]. Let G be the Galois group of a tamely ramified extension F/Q all of whose decomposition groups are abelian.…”