The group Gal(k3(2)|k) for k = ℚ(−3,d) of type (3,3)

Azizi, Abdelmalek; Talbi, Mohamed; Derhem, A.; Mayer, Daniel C.

doi:10.1142/s1793042116501207

Cited by 2 publications

(2 citation statements)

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“…Since all second 3-class groups M = G 2 3 k in Theorem 8.3 and Theorem 8.7 of [7] have relation rank d 2 M = 5, they cannot coincide with the 3-tower group G = G ∞ 3 k, and the corresponding 3-class field tower must have length 3 k at least 3 whenever the coclass is cc(M) ≥ 2.…”

Section: Simple and Advanced Tree Topologiesmentioning

confidence: 99%

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Recent progress in determining p-class field towers

Mayer¹

2016

GJOM

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For a fixed prime p, the p-class tower Fp∞K of a number field K is considered to be known if a pro-p presentation of the Galois group G = Gal (Fp∞K ∕ K) is given. In the last few years, it turned out that the Artin pattern AP(K) = (τ(K), κ(K)) consisting of targets τ(K) = (ClpL) and kernels κ(K) = (ker JL|K) of class extensions JL|K: Clp K → ClpL to unramified abelian subfields L|K of the Hilbert p-class field Fp1K only suffices for determining the two-stage approximation M = G ∕ G'' of G. Additional techniques had to be developed for identifying the group G itself: searching strategies in descendant trees of finite p-groups, iterated and multilayered IPADs of second order, and the cohomological concept of Shafarevich covers involving relation ranks. This enabled the discovery of three-stage towers of p-class fields over quadratic base fields K = ℚ(√ d) for p ∈ {2, 3, 5}. These non-metabelian towers reveal the new phenomenon of various tree topologies expressing the mutual location of the groups G and M.

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Section: Simple and Advanced Tree Topologiesmentioning

confidence: 99%

“…A diagram of the pruned descendant tree T * R with root R = 25, 2 , where the finite 5-groups M and H j are located, is shown in [46, § 7, Fig.1]. Polycyclic power commutator presentations of the groups H j are given in[46, § 7] and a diagram of their normal lattice, including the lower and upper central series, is drawn in[46,§ 7, Fig.…”

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confidence: 99%