THE SECOND p-CLASS GROUP OF A NUMBER FIELD

Abstract

Abstract. For a prime p ≥ 2 and a number field K with p-class group of type (p, p) it is shown that the class, coclass, and further invariants of the metabelian Galois group G = Gal(F 2 p (K)|K) of the second Hilbert p-class field F 2 p (K) of K are determined by the p-class numbers of the unramified cyclic extensions N i |K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field K = Q( √ D) and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields L i |Q of absolute degree p of the dihedral fields N i . As an application, the structure of the automorphism group G = Gal(F 2 3 (K)|K) of the second Hilbert 3-class field F 2 3 (K) is analysed for all quadratic fields K with discriminant −10 6 < D < 10 7 and 3-class group of type (3, 3) by computing their principalisation types. The distribution of these metabelian 3-groups G on the coclass graphs G(3, r), 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated. IntroductionFor an algebraic number field K and a prime p ≥ 2 we denote by Cl p (K) the p-class group of K, that is the Sylow p-subgroup Syl p Cl(K) of its ideal class group. In this paper we shall be concerned with number fields having an elementary abelian bicyclic p-class group of type (p, p). We define the p-class field tower of The aim of the present paper is to investigate the second p-class group G = Gal(F 2 p (K)|K), that is the Galois group of the two-stage tower,, of a base field K with p-class group of type (p, p). The second p-class group G is distinguished by the special property that its commutator subgroupis isomorphic to the p-class group of the first Hilbert p-class field of K, whence G is metabelian.The theory of second p-class groups was initiated by Scholz and Taussky [38,39], using Schreier's concept of group extensions [36,37] Eick, et al. [11,10,12].

The distribution of second p-class groups on coclass graphs

“…Malle. The heuristic is in good accordance with our extensive computational results for quadratic base fields in [58]. Our results have in fact actually been used in [12, pp.…”

“…These TKTs define infinite sequences, in fact periodic coclass families ( § 2), of possible groups G 2 3 (K), and neither Heider and Schmithals [45] nor Brink and Gold [24,25] have been aware that the TTT is able to identify a unique member of the sequences, as we proved in [58,60]. Theorem 1.3.…”

“…They were communicated in [45, p. 19] (see also [11, p. 165]). Among the 2020 complex quadratic fields K = Q( √ D) with discriminants −10 6 < D < 0 and Cl 3 (K) of type (3,3), there are 391 cases (19.4%) having one of the two pairs of TKT and TTT in Theorem 1.6 [58,Tbl. 3,p.…”

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]).…”

“…In den letzten Jahren haben auch Bush und vor allem D. Mayer [188,189,190,191,49] die Kapitulation in unverzweigten kubischen Erweiterungen und die dazugehöri-gen Klassenkörpertürme ausgiebig untersucht.…”