PRINCIPALISATION ALGORITHM VIA CLASS GROUP STRUCTURE DANIEL C. MAYER Abstract. For an algebraic number field K with 3-class group Cl3(K) of type (3; 3) the structure of the 3-class groups Cl3(Ni) of the four unramified cyclic cubic extension fields Ni (1 ≤ i ≤ 4) of K is calculated with the aid of presentations for the metabelian Galois group G = Gal(F2(K)jK) of the second Hilbert 3-class field F2(K) of K. In the case of a quadratic 3 p 3 base field K = Q( D) it is shown that the structure of the 3-class groups of the four S3-fields N1;:::;N4 determines the type of the principalisation of the 3-class group of K in N1;:::;N4. This provides an alternative to the classical principalisation algorithm by Scholz and Taussky. The new algorithm, which is easily automatisable and executes very quickly, is implemented in PARI/GP and applied to all 4 596 quadratic fields with discriminant −106 < D < 107 and 3-class group of type (3; 3) to obtain extensive statistics of their principalisation types and the distribution of their second 3-class groups G on the coclass graphs G(3; r), 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green. 1. Introduction The principal ideal theorem, which has been conjectured by Hilbert in 1898 [14, p.14], states that every ideal of a number field K becomes principal when it is extended to the Hilbert class field F1(K) of K, that is the maximal abelian unramified extension field of K. Inspired by the Artin-Furtw¨anglerproof [2, 14] of the principal ideal theorem, Scholz and Taussky investigated 1 the principalisation in intermediate fields K < N < F3(K) between a base field K with non-cyclic 1 3-class group of type (3; 3) and its Hilbert 3-class field F3(K). They developed an algorithm to compute the principalisation of K in its four unramified cyclic cubic extension fields N1;:::;N4 for the case of a complex quadratic base field K [29]. This algorithm is probabilistic, since it decides if an ideal a of K becomes principal in Ni for some 1 ≤ i ≤ 4 by testing local cubic residue characters of a principal ideal cube (α) = a3, associated with the ideal a, and of the fundamental unit "i of the non-Galois cubic subfield Li of the complex S3-field Ni with respect to a series of rational test primes (p`)`≥1 and terminating when a critical test prime occurs [20, p.81, Step 8]. An upper bound for the minimal critical test prime p`0 cannot be given effectively. It can only be estimated by means of Chebotar¨ev'sdensity theorem [29, 16], thus causing uncertainty. An entirely different approach to the principalisation problem will be presented in this paper. It is based on a purely group theoretical connection between the structure of the abelianisations Mi/γ2(Mi) of the four maximal normal subgroups Mi of an arbitrary metabelian 3-group G with G/γ2(G) of type (3; 3) and the kernels Ker(Vi) of the transfers Vi from G to Mi (1 ≤ i ≤ 4). By the Artin reciprocity law of class field theory [1], a corresponding number theoretical connection is established between the structure of the 3-class groups Cl3(Ni) of the four unramified cyclic cubic extension fields Ni of an arbitrary base field K with 3-class group Cl3(K) of type (3; 3) and the principalisation kernels Ker(jNijK ) of the class extension homomorphisms jNijK : Cl3(K) −! Cl3(Ni) (1 ≤ i ≤ 4), if the group theoretical statements are applied to the second 3-class group 2 2 G = Gal(F3(K)jK) of K [22], that is the Galois group of the second Hilbert 3-class field F3(K) = 1 1 F3(F3(K)) of K. 1 2 We begin by comparing the big two-stage tower K < F3(K) < F3(K) and the four little two- 1 1 stage towers K < F3(K) < F3(Ni) (1 ≤ i ≤ 4) of 3-class fields in section 2.1. Based on these Date: January 01, 2011. 2000 Mathematics Subject Classification. Primary 11R29,11R11,11R20; Secondary 20D15. Key words and phrases. 3-class groups, principalisation of 3-classes, quadratic base fields, number fields with S3-group, metabelian 3-groups, coclass graphs. Research supported by the Austrian Science Fund, Grant Nr. J0497-PHY. 1 2 DANIEL C. MAYER relationships, section 2.2 is devoted to the proof that the 3-class groups Cl3(Ni) (1 ≤ i ≤ 4) have the structure of a nearly homocyclic abelian 3-group of 3-rank two, provided that the index of 2 nilpotency of the second 3-class group G = Gal(F3(K)jK) is not too small. The structure of the remaining 3-class groups is determined in section 2.3.1, if G is of coclass cc(G) = 1. However, the central results of this paper concern groups G of coclass cc(G) ≥ 2 and are developed in section 2.3.2. They establish the theoretical background for the new principalisation algorithm in section 3 with the aid of the number " of elementary abelian 3-groups of type (3; 3; 3) and of 3-rank three among the 3-class groups Cl3(Ni) (1 ≤ i ≤ 4). 2. The structure of the four 3-class groups Cl3(Ni) 2.1. Two-stage towers of 3-class fields. Let K be an algebraic number field with elementary abelian 3-class group Cl3(K) of type (3; 3). Then, by class field theory, there exist four unramified 1 cyclic cubic extension fields N1;:::;N4 of K within the first Hilbert 3-class field F3(K) of K. 1 1 Consequently, the first Hilbert 3-class field F3(Ni) of Ni is an intermediate field between F3(K) 2 and the second Hilbert 3-class field F3(K) of K, for 1 ≤ i ≤ 4. 1 Since F3(K) is the maximal abelian unramified extension field of K with a power of 3 as rela- 2 1 tive degree, the Galois groups G = Gal(F3(K)jK) of the big two-stage tower of K, K < F3(K) ≤ 2 1 1 1 F3(K), and Γi = Gal(F3(Ni)jK) of the four little two-stage towers of K, K < F3(K) ≤ F3(Ni) 1 with 1 ≤ i ≤ 4, are non-abelian, except in the degenerate case of a single-stage tower F3(K) = 1 2 2 1 F3(Ni) = F3(K). Since their commutator subgroups are given by γ2(G) = Gal(F3(K)jF3(K)) ' 1 1 1 1 Cl3(F3(K)) and γ2(Γi) = Gal(F3(Ni)jF3(K)) < Ai = Gal(F3(Ni)jNi) ' Cl3(Ni) by [1], G and Γi 1 are metabelian 3-groups with abelianisations G/γ2(G) and Γi/γ2(Γi) isomorphic to Gal(F3(K)jK) ' Cl3(K) and thus of type (3; 3). In the following section 2.2, we determine the standard structure of the abelian maximal normal subgroup Ai ' Cl3(Ni) of Γi with 1 ≤ i ≤ 4 for a given second 3-class group G of K. Since the four maximal normal subgroups M1;:::;M4 of G are associated with the extensions N1;:::;N4 2 by Mi = Gal(F3(K)jNi), their commutator groups and abelianisations are given by γ2(Mi) = 2 1 1 Gal(F3(K)jF3(Ni)) and Mi/γ2(Mi) = Gal(F3(Ni)jNi) ' Cl3(Ni). Since γ2(Mi) is a characteristic subgroup of Mi, it is a normal subgroup of G. Consequently, we have the relation G/γ2(Mi) = 2 2 1 1 Gal(F3(K)jK)=Gal(F3(K)jF3(Ni)) ' Gal(F3(Ni)jK) = Γi and the connection between Γi and G on the one hand, and Ai and Mi on the other hand, can be summarised as follows: Γ ' G/γ (M ) ; (1) i 2 i Ai ' Mi/γ2(Mi) : 2.2. Nearly homocyclic 3-class groups of type (3q+r; 3q). The concept of nearly homocyclic p-groups with an arbitrary prime p ≥ 2 appears in [8, p.68, Th.3.4] and is treated systematically in [25, 2.4]. For our purpose, it suffices to consider the special case p = 3. Definition 2.1. By the nearly homocyclic abelian 3-group A(3; n) of order 3n, for an integer n ≥ 2, we understand the abelian group of type (3q+r; 3q), where n = 2q + r with integers q ≥ 1 and 0 ≤ r < 2. Additionally, we define that A(3; 1) denotes the cyclic group C(3) of order 3 and A(3; 0) the trivial group 1. The application of Blackburn's well-known theorem 3.4 in [8, p.68] to the Galois groups Γi = 1 1 1 Gal(F3(Ni)jK) of the four little two-stage towers K < F3(K) ≤ F3(Ni) with abelian maximal 1 normal subgroups Ai = Gal(F3(Ni)jNi) ' Cl3(Ni) will show that in general the 3-class groups u v Cl3(Ni) (1 ≤ i ≤ 4) are nearly homocyclic abelian 3-groups A(3; u + v) of type (3 ; 3 ) with 1 ≤ v ≤ u ≤ v + 1. The phrase `in general' is made precise in the following theorems, in which we 2 distinguish second 3-class groups G = Gal(F3(K)jK) of coclass cc(G) = 1 and of coclass cc(G) ≥ 2 and use concepts and notation of our papers [22, 23]. 2.2.1. Second 3-class groups G of coclass cc(G) = 1. Let G be a metabelian 3-group of order jGj = 3n and nilpotency class cl(G) = m − 1, where n = m ≥ 3.