Computation of Galois groups associated to the 2-class towers of some quadratic fields

Abstract

The p-group generation algorithm from computational group theory is used to obtain information about large quotients of the pro-2 group G = Gal (k nr,2 /k) for k = Q( √ d) with d = −445, −1015, −1595, −2379. In each case we are able to narrow the identity of G down to one of a finite number of explicitly given finite groups. From this follow several results regarding the corresponding 2-class tower.

Der Briefwechsel Hasse - Scholz - Taussky

“…Remmert [268] [14], [215]), Bartholdi [16], Bush [47,48,49], Hajir [107,108], Kuhnt [160], Maire [181,182], D. Mayer [188,189,190,191]), McLeman [193], Nover [231], Steurer [307]. Auch die gruppentheoretische Seite dieses Themenkomplexes wurde ausgiebig untersucht (Magnus [180], Serre [293], Nebelung, [222]).…”

Index-<i>p</i> Abelianization Data of <i>p</i>-Class Tower Groups

“…However, there also exists another extension of the IPAD which is not covered by the TTT. It has also been used already in previous investigations by Boston, Bush and Nover [12] [14] [15] and is constructed from the usual IPAD ( ) ( )…”

Maximal unramified 3-extensions of imaginary quadratic fields and SL2(Z3)

“…In [4] and [5] the p-group generation algorithm is used to compute the Galois groups of several p-extensions with restricted ramification. Here we use it to verify that Gal(k nr,3 /k) ∼ = G 1 for several different imaginary quadratic fields k. For the reader's convenience we recall some definitions and give a brief description of the method.…”

“…This information is sometimes sufficient to eliminate all but finitely many groups from the tree of descendants described above, in which case we are left with a finite number candidates for the Galois group. A more precise formulation of the method and several examples in the case p = 2 can be found in [5].…”

“…Indeed to date the largest length observed is 3 and in all these examples p = 2, see [5]. In the next section we will define a family of finite Schur-σ groups with p = 3 and then show that the derived length for groups in this family is unbounded.…”