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Recent progress in determining p-class field towers | Daniel C. Mayer
; | Date: |
31 May 2016 | Abstract: | For a fixed prime p, the p-class tower F(p,infinity,K) of a number field K is
considered to be known if a pro-p presentation of the Galois group H = Gal(
F(p,infinity,K)/K ) is given. In the last few years, it turned out that the
Artin pattern AP(K) = (tau(K),kappa(K)) consisting of targets tau(K) =
(Cl(p,L)) and kernels kappa(K) = (ker(J(L/K)) of class extensions J(L/K):
Cl(p,K) --> Cl(p,L) to unramified abelian subfields L/K of the Hilbert p-class
field F(p,1,K) only suffices for determining the two-stage approximation G =
H/H" of H. Additional techniques had to be developed for identifying the group
H 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(
squareroot(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 H and G. | Source: | arXiv, 1605.9617 | Services: | Forum | Review | PDF | Favorites |
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