Principalisation Algorithm Via Class Group Structure

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)|K) of the second Hilbert 3-class field F2(K) of K. In the case of a quadratic 3 √ 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-Furtwänglerproof [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 εi 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 Chebotarëv’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(jNi|K ) of the class extension homomorphisms jNi|K : 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)|K) 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)|K) 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 ε 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)|K) of the big two-stage tower of K, K < F3(K) ≤ 2 1 1 1 F3(K), and Γi = Gal(F3(Ni)|K) 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)|F3(K)) ' 1 1 1 1 Cl3(F3(K)) and γ2(Γi) = Gal(F3(Ni)|F3(K)) < Ai = Gal(F3(Ni)|Ni) ' 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)|K) ' 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)|Ni), their commutator groups and abelianisations are given by γ2(Mi) = 2 1 1 Gal(F3(K)|F3(Ni)) and Mi/γ2(Mi) = Gal(F3(Ni)|Ni) ' 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)|K)/Gal(F3(K)|F3(Ni)) ' Gal(F3(Ni)|K) = 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)|K) of the four little two-stage towers K < F3(K) ≤ F3(Ni) with abelian maximal 1 normal subgroups Ai = Gal(F3(Ni)|Ni) ' 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)|K) 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 |G| = 3n and nilpotency class cl(G) = m − 1, where n = m ≥ 3. Then G is of coclass cc(G) = PRINCIPALISATION ALGORITHM 3 n − cl(G) = 1 and the commutator factor group G/γ2(G) of G is of type (3, 3) [8, 24]. The lower central series of G is deﬁned recursively by γ1(G) = G and γj(G) = [γj−1(G),G] for j ≥ 2. The centraliser χ2(G) = {g ∈ G | [g, u] ∈ γ4(G) for all u ∈ γ2(G)} of the two-step factor group γ2(G)/γ4(G), that is,

χ2(G)/γ4(G) = CentraliserG/γ4(G)(γ2(G)/γ4(G)) , is the biggest subgroup of G such that [χ2(G), γ2(G)] ≤ γ4(G). It is characteristic, contains the commutator group γ2(G), and coincides with G, if and only if m = 3. Let the isomorphism invariant k = k(G) of G be deﬁned by

[χ2(G), γ2(G)] = γm−k(G) , where k = 0 for 3 ≤ m ≤ 4, and 0 ≤ k ≤ 1 for m ≥ 5, according to Miech [24, p.331]. Suppose that generators of G = hx, yi are selected such that x ∈ G \ χ2(G), if m ≥ 4, and y ∈ χ2(G)\γ2(G). We deﬁne the main commutator s2 = [y, x] ∈ γ2(G) and the higher commutators x−1 sj = [sj−1, x] = sj−1 ∈ γj(G) for j ≥ 3. Then G satisﬁes two relations for third powers of the generators x and y of G,

3 w 3 3 z (2) x = sm−1 and y s2s3 = sm−1 with exponents − 1 ≤ w, z ≤ 1 , according to Miech [24, p.332, Th.2, (3)]. Blackburn uses the notation δ = w and γ = z for these relational exponents [8, p.84, (36),(37)]. Additionally, the group G satisﬁes relations for third powers of the higher commutators, 3 3 sj+1sj+2sj+3 = 1 for 1 ≤ j ≤ m − 2 , and the main commutator relation of Miech [24, p.332, Th.2, (2)],

a (3) [y, s2] = sm−1 ∈ [χ2(G), γ2(G)] = γm−k(G) , with exponent −1 ≤ a ≤ 1. Blackburn uses the notation β = a [8, p.82, (33)]. (m) By Ga (z, w) we denote the representative of an isomorphism class of metabelian 3-groups G of coclass cc(G) = 1 and of order |G| = 3m, which satisfies the relations (3) and (2) with a fixed system of exponents a, w, and z. The maximal normal subgroups Mi of G contain the commutator group γ2(G) of G as a normal subgroup of index 3 and thus are of the shape Mi = hgi, γ2(G)i. We define an ordering by −1 g1 = y, g2 = x, g3 = xy and g4 = xy . The commutator groups γ2(Mi) are of the general form gi−1 γ2(Mi) = hs2, . . . , sm−1i , according to [22, Cor.3.1.1], and in particular

( 1, if k = 0, γ2(M1) = (4) γm−1(G), if k = 1, γ2(Mi) = γ3(G) for 2 ≤ i ≤ 4.

Theorem 2.1. The structure of the 3-class groups Cl3(Ni) of the four unramified cyclic cubic extension fields N1,...,N4 of an arbitrary base field K having a 3-class group Cl3(K) of type 2 n (3, 3) and a second 3-class group G = Gal(F3(K)|K) of coclass cc(G) = 1, order |G| = 3 , and class cl(G) = m−1, where 3 ≤ m = n, is given by the following nearly homocyclic abelian 3-groups.

( A(3, m − 1), if [χ2(G), γ2(G)] = 1, k = 0, m ≥ 5 , (5) Cl3(N1) ' A(3, m − 2), if [χ2(G), γ2(G)] = γm−1(G), k = 1, m ≥ 6 ,

(6) Cl3(Ni) ' A(3, 2) for 2 ≤ i ≤ 4, if m ≥ 4 . 1 Proof. It is essential that the metabelian 3-groups Γi = Gal(F3(Ni)|K) are of coclass cc(G) = 1 [16, p.9, Kor.]. We begin by investigating the distinguished extension N1. 4 DANIEL C. MAYER

Suppose ﬁrst that [χ2(G), γ2(G)] = 1, that is k = 0. Then γ2(M1) = 1, by [22, Cor.3.1.1], the m group Γ1 is isomorphic to G, by formula (1), and has the order |Γ1| = 3 . For m ≥ 5, the abelian normal subgroup A1 of Γ1 is isomorphic to A(3, m − 1), according to Blackburn [8, Thm.3.4], where the bound p + 2 = 5 for the index of nilpotency m is due to the specialisation p = 3. Next we consider the case [χ2(G), γ2(G)] = γm−1(G), that is k = 1. Then γ2(M1) = γm−1(G), by [22, Cor.3.1.1], the group Γ1 is isomorphic to G/γm−1(G) ' G/ζ1(G), by formula (1), and is the immediate predecessor of G on the coclass graph G(3, 1) [12], thus being isomorphic to (m−1) m−1 G0 (0, 0) [25, 23] and of order |Γ1| = 3 . For m ≥ 6, it follows that A1 ' A(3, m − 2), according to [8]. However, Blackburn’s result cannot be applied to the other three extensions Ni with 2 ≤ i ≤ 4, µ since the three isomorphic groups Γi ' G/γ3(G) [22, Cor.3.1.1] are of order |Γi| = 3 with exponent µ = 3 < 5. Therefore we must determine the structure of the abelian normal subgroup Ai of Γi by the following consideration. In the case m ≥ 4, the group γ3(G) > 1 is non-trivial. Since Γi is a predecessor of G on the coclass graph G(3, 1), it can only be isomorphic to the extra (3) special 3-group G0 (0, 0) of exponent 3 [25, 23], whose four maximal normal subgroups are all of type (3, 3) and are thus isomorphic to A(3, 2). For G of coclass cc(G) = 1, it remains to investigate the structure of the following 3-class groups: m−1 3 • Cl3(N1) of order 3 = 3 for m = 4 with [χ2(G), γ2(G)] = 1, k = 0, m−2 3 • Cl3(N1) of order 3 = 3 for m = 5 and [χ2(G), γ2(G)] = γ4(G) > 1, k = 1, 2 • all four Cl3(Ni) with 1 ≤ i ≤ 4 of order 3 for m = 3.

2.2.2. Second 3-class groups G of coclass cc(G) ≥ 2. The metabelian 3-groups G of coclass cc(G) ≥ 2, cannot be CF-groups, since they must have at least one bicyclic factor γ3(G)/γ4(G). Similarly as in section 2 of [22], we declare an isomorphism invariant e = e(G) of G by e + 1 = min{3 ≤ j ≤ m | 1 ≤ |γj(G)/γj+1(G)| ≤ 3}. This invariant 2 ≤ e ≤ m − 1 characterises the first cyclic factor γe+1(G)/γe+2(G) of the lower central series of G, except γ2(G)/γ3(G), which is always cyclic. e can be calculated from the 3-exponent n of the order |G| = 3n and the class cl(G) = m − 1, resp. the index m, of nilpotency of G by the formula e = n − m + 2. Since the coclass of G is given by cc(G) = n − cl(G) = n − m + 1, we have the relation e = cc(G) + 1. For a group G of coclass cc(G) ≥ 2 we need a generalisation of the group χ2(G). Denoting by m the index of nilpotency of G, we let χj(G) with 2 ≤ j ≤ m − 1 be the centralisers of two-step factor groups γj(G)/γj+2(G) of the lower central series, that is, the biggest subgroups of G with the property [χj(G), γj(G)] ≤ γj+2(G). They form an ascending chain of characteristic subgroups of G, γ2(G) ≤ χ2(G) ≤ ... ≤ χm−2(G) < χm−1(G) = G, which contain the commutator group γ2(G). χj(G) coincides with G, if and only if j ≥ m − 1. Similarly as in section 2 of [22], we characterise the smallest two-step centraliser different from the commutator group by an isomorphism invariant s = s(G) = min{2 ≤ j ≤ m − 1 | χj(G) > γ2(G)}. The following assumptions for a metabelian 3-group G of coclass cc(G) ≥ 2 with abelianisation G/γ2(G) of type (3, 3) can always be satisfied, according to [25]. Let G be a metabelian 3-group of coclass cc(G) ≥ 2 with abelianisation G/γ2(G) of type (3, 3). Assume that G has order |G| = 3n, class cl(G) = m − 1, and invariant e = n − m + 2 ≥ 3, where 3 3 4 ≤ m < n ≤ 2m − 3. Let generators of G = hx, yi be selected such that γ3(G) = hy , x , γ4(G)i, x ∈ G \ χs(G), if s < m − 1, and y ∈ χs(G) \ γ2(G). Suppose that the ordering of the four maximal normal subgroups of G is defined by Mi = hgi, γ2(G)i with g1 = y, g2 = x, g3 = xy, −1 and g4 = xy . Let the main commutator of G be declared by s2 = t2 = [y, x] ∈ γ2(G) and higher commutators recursively by sj = [sj−1, x], tj = [tj−1, y] ∈ γj(G) for j ≥ 3. Starting with 3 3 the powers σ3 = y , τ3 = x ∈ γ3(G), let σj = [σj−1, x], τj = [τj−1, y] ∈ γj(G) for j ≥ 4. With exponents −1 ≤ α, β, γ, δ, ρ ≤ 1, let the following relations be satisfied

3 −ρβ −1 ρβ γ δ −1 ρδ α β −ρ (7) s2 = σ4σm−1τ4 , s3σ3σ4 = σm−2σm−1τe , t3 τ3τ4 = σm−2σm−1τe , τe+1 = σm−1.

Finally, let [χs(G), γe(G)] = γm−k(G) with 0 ≤ k ≤ 1. PRINCIPALISATION ALGORITHM 5

Theorem 2.2. The structure of the 3-class groups Cl3(Ni) of the four unramified cyclic cubic extension fields N1,...,N4 of an arbitrary base field K having a 3-class group Cl3(K) of type 2 n (3, 3) and a second 3-class group G = Gal(F3(K)|K) of coclass cc(G) ≥ 2, order |G| = 3 , class cl(G) = m, and invariant e = n − m + 2 ≥ 3, where 4 ≤ m < n ≤ 2m − 3, is given by the following nearly homocyclic abelian 3-groups.

( A(3, m − 1), if [χs(G), γe(G)] = 1, k = 0, m ≥ 5 , (8) Cl3(N1) ' A(3, m − 2), if [χs(G), γe(G)] = γm−1(G), k = 1, m ≥ 6 ,

(9) Cl3(N2) ' A(3, e) for e ≥ 4 , (4) (10) Cl3(Ni) ' A(3, 3) for 3 ≤ i ≤ 4, if Γi 6' G0 (1, 0) ' Syl3A9 . Proof. For each 1 ≤ i ≤ 4, equation (1) speciﬁes the order of the Galois group of the ith little two- n stage tower as |Γi| = |G|/|γ2(Mi)|, where |G| = 3 by assumption. According to [22, Cor.4.1.1], the orders of the commutator groups of the Mi are

 e−2 3 for i = 1, [χs(G), γe(G)] = 1, k = 0,  e−1 3 for i = 1, [χs(G), γe(G)] = γm−1(G), k = 1, |γ2(Mi)| = m−3 3 for i = 2,  3n−4 for 3 ≤ i ≤ 4.

Using the relation e = n − m + 2, we obtain

 m 3 for i = 1, [χs(G), γe(G)] = 1,  m−1 3 for i = 1, [χs(G), γe(G)] = γm−1(G), |Γi| = e+1 3 for i = 2,  34 for 3 ≤ i ≤ 4.

Since Γi is a metabelian 3-group of coclass cc(G) = 1 [16, p.9, Kor.], the structure of the abelian maximal normal subgroup Ai of Γi is given by

( A(3, m − 1), if [χs(G), γe(G)] = 1, k = 0, m ≥ 5, A1 ' A(3, m − 2), if [χs(G), γe(G)] = γm−1(G), k = 1, m − 1 ≥ 5,

A2 ' A(3, e), for e + 1 ≥ 5, (4) Ai ' A(3, 3) for 3 ≤ i ≤ 4, if Γi 6' G0 (1, 0) ' Syl3A9, according to [8, Th.3.4], for 1 ≤ i ≤ 2, and by an immediate analysis of the four isomorphism classes of metabelian 3-groups of order 3m with index of nilpotency m = 4 [23], for 3 ≤ i ≤ 4. Rep- (4) (4) (4) resentatives for these four isomorphism classes are three groups G0 (0, 0), G0 (0, 1), G0 (−1, 0), whose abelian maximal normal subgroup is nearly homocyclic of type (9, 3), and the exceptional (4) group G0 (1, 0) ' Syl3A9, which has the elementary abelian 3-group of type (3, 3, 3) as its abelian maximal normal subgroup.

For G of coclass cc(G) ≥ 2, it remains to investigate the structure of the following 3-class groups: m−k−1 3 • Cl3(N1) of order 3 = 3 for small values of the index of nilpotency m, namely m = 4, if [χs(G), γe(G)] = 1, k = 0, and m = 5, if [χs(G), γe(G)] = γm−1(G), k = 1, e 3 • Cl3(N2) of order 3 = 3 for groups G of coclass cc(G) = 2 with e = 3, 3 • Cl3(Ni) (3 ≤ i ≤ 4) of order 3 without restrictions for the parameters m ≥ 4 and e ≥ 3. In the following table 1, we give an overview of all systems (m, n, e, k) of parameters of the second 3-class group G of coclass cc(G) ≥ 2, which have to be analysed for the 3-class groups Cl3(Ni) (1 ≤ i ≤ 4) marked by the symbol ×. 6 DANIEL C. MAYER

Table 1. 3-class groups to be investigated for certain parameters m, n, e, k of G

m n e k Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) 4 5 3 0 × × × × 5 6 3 1 × × × × 5 6 3 0 × × × ≥ 6 ≥ 7 3 ≥ 0 × × × 5 7 4 0 × × ≥ 6 ≥ 8 ≥ 4 ≥ 0 × ×

2.3. Elementary abelian 3-class groups of type (3, 3, 3). Several authors, namely Scholz [28, p.218], Kisilevsky [18, p.205, Thm.3], Heider and Schmithals [16, p.11, Satz 7], and Brink [9, pp.51–52], have pointed out the theoretical possibility that the elementary abelian 3-group of order 27 can occur as the 3-class group Cl3(N) of an unramified cyclic cubic extension field N of a base field K with 3-class group of type (3, 3). The most explicit result among these statements is due to Heider and Schmithals. They prove that the occurence of an elementary abelian 3-class group Cl3(N) of order 27 is restricted to extensions N|K satisfying condition (B) in the sense of Taussky [31], that is, having a partial principalisation without fixed point [23]. Since the nearly homocyclic abelian 3-groups A(3, n) with n ≥ 2, which generally occur as Cl3(N), according to the theorems 2.1 and 2.2, are of 3-rank two, the elementary abelian 3-group of type (3, 3, 3) is the unique possiblity for Cl3(N) to be of 3-rank three. Unfortunately, it was impossible to find a numerical example, let alone a general criterion, for the occurrence of 3-class groups of type (3, 3, 3) in the bibliography, up to now. At first it was completely unknown, if 3-class groups of type (3, 3, 3) exist at all, if they appear sporadically or stochastically, or if their occurrence is ruled by deterministic laws. In the present paper we systematically analyse this question by means of presentations for the 2 second 3-class group G = Gal(F3(K)|K), which have been given by Blackburn [8, pp.82–84] in the case of coclass cc(G) = 1 and by Nebelung [25, p.94, Satz 3.4.5] in the case of coclass cc(G) ≥ 2, and arrive at the surprising result that the number ε of elementary abelian 3-groups of type (3, 3, 3) among the 3-class groups Cl3(Ni) is connected with the principalisation type κ of 3-classes of K in Ni for 1 ≤ i ≤ 4 by strict rules. These connections offer new algorithmic√ possibilities for computing the principalisation type of 3-classes of a quadratic base field K = Q( D) with 3-class group of type (3, 3), independent from the classical algorithm for complex quadratic fields by Scholz and Taussky [29] and its modification for real quadratic fields by Heider and Schmithals [16]. The new algorithm is based on determining the structure of the 3-class groups Cl3(N) of the unramified cyclic cubic extensions N|K, that is, of four S3-fields N of absolute degree six. With the aid of an implementation of this algorithm in the number theoretical computer algebra system PARI/GP [27], we have computed the principalisation type of the 3-classes of all 4 596 quadratic base fields K with 3-class group of type (3, 3) and discriminant −106 < D < 107. The 2 resulting extensive statistics of principalisation types and second 3-class groups G = Gal(F3(K)|K) will be discussed in section 4. 2.3.1. Second 3-class groups G of coclass cc(G) = 1.

Theorem 2.3. The structure of the 3-class groups Cl3(Ni) of the four unramiﬁed cyclic cubic extensions Ni (1 ≤ i ≤ 4) of an arbitrary base ﬁeld K with 3-class group Cl3(K) of type (3, 3) and 2 m second 3-class group G = Gal(F3(K)|K) of coclass cc(G) = 1 and order |G| = 3 ist given by the following array for small indices of nilpotency 3 ≤ cl(G) + 1 = m ≤ 5. In the case m = 3, suppose that the generating element y of the maximal normal subgroup M1 = 2 Gal(F3(K)|N1) = hy, s2i of G has order 3. PRINCIPALISATION ALGORITHM 7

 A(3, 2), if m = 3,  4 A(3, 3), if m = 4,G 6' G0(1, 0) ' Syl3A9, (11) Cl3(N1) ' 4 C(3) × C(3) × C(3), if m = 4,G ' G0(1, 0) ' Syl3A9,  A(3, 3), if k = 1, m = 5,

( 3 A(3, 2), if G ' G0(0, 0), (12) Cl3(Ni) ' 3 for 2 ≤ i ≤ 4, if m = 3. C(9), if G ' G0(0, 1), Proof. We make use of the well-known properties of the six isomorphism classes of metabelian (m) 3-groups Ga (z, w) of coclass cc(G) = 1 with the smallest nilpotency classes 3 ≤ m ≤ 4 [23]. We start with the distinguished extension N1. First, let [χ2(G), γ2(G)] = 1, that is k = 0. Then the group Γ1 is isomorphic to G and is of m order |Γ1| = 3 , according to equation (1) and [22, Cor.3.1.1]. For m = 4, the abelian maximal normal subgroup A1 ' M1 is either nearly homocyclic of type (9, 3) and thus isomorphic to A(3, 3), if G belongs to one of the three isomorphism classes (4) (4) (4) G0 (0, 0), G0 (0, 1), G0 (−1, 0), or elementary abelian of type (3, 3, 3), if G is isomorphic to (4) G0 (1, 0), the 3-Sylow subgroup Syl3A9 of the alternating group of degree 9. However, for m = 3 the structure of the abelian maximal normal subgroup A1 ' M1 can only be nearly homocyclic of type (3, 3) and thus isomorphic to A(3, 2), in the case of the extra special (3) group G ' G0 (0, 0) of exponent 3 with four abelian maximal normal subgroups of type (3, 3), (3) as well as in the case of the extra special group G ' G0 (0, 1) of exponent 9 with one abelian maximal normal subgroup A1 ' M1 = hy, s2i of type (3, 3), which is distinguished by our choice of the generating element y, and three further cyclic maximal normal subgroups of order 9. Now we consider [χ2(G), γ2(G)] = γm−1(G), that is k = 1, with m = 5 and thus γ4(G) > 1. According to equation (1) and [22, Cor.3.1.1], the group Γ1 ' G/γ4(G) ' G/ζ1(G) is the (4) immediate predecessor of G on the coclass graph G(3, 1) and thus is isomorphic to G0 (0, 0) with the unique abelian maximal normal subgroup A1 ' A(3, 3). For the other extensions Ni with 2 ≤ i ≤ 4, all three groups Γi ' G/γ3(G) ' G [22, Cor.3.1.1] coincide with G, since γ3(G) = 1 is trivial for m = 3. The structure of the abelian maximal normal subgroup Ai ' Mi is either of type (3, 3) and thus isomorphic to A(3, 2), if G is isomorphic to (3) i−2 G0 (0, 0), or cyclic of order 9, taking into account that Mi = hxy i, if G is isomorphic to (3) G0 (0, 1).

Corollary 2.3.1. The following table gives the structure of the 3-class groups Cl3(Ni) in dependence on the principalisation type κ [23].

2.3.2. Second 3-class groups G of coclass cc(G) ≥ 2. The basic idea for deciding if a 3-class group 3 Cl3(Ni) ' Mi/γ2(Mi), whose order has been determined as 3 , is nearly homocyclic of type (9, 3) or elementary abelian of type (3, 3, 3), consists in an estimate of the order of the cosets of the generators gi, s2, σ3, . . . , σm−1, τ3, . . . , τe of Mi = hgi, γ2(G)i with respect to the commutator group γ2(Mi). If these orders are all bounded from above by 3, then we have an elementary abelian 3-class group of type (3, 3, 3), otherwise a nearly homocyclic 3-class group of type (9, 3). First we summarise general facts concerning the columns of table 1 in the following three sections.

2.3.3. The 3-class group Cl3(N1) ' M1/γ2(M1).

M1 = hy, γ2(G)i = hy, s2, σ3, . . . , σm−1, τ3, . . . , τei,

γ2(M1) = ht3, τ4, . . . , τe+1i.

Since τ4, . . . , τe ∈ γ2(M1) for e ≥ 4, the order of the cosets τ4,..., τe equals 1. 3 3 The relation τ3 τ4 τ5 = 1 for third powers implies ord(τ3) ≤ 3. 8 DANIEL C. MAYER

Table 2. 3-class groups for G of coclass cc(G) = 1 and index of nilpotency 3 ≤ m ≤ 5

1 m k Type κ Cl3(F3(K)) Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) ε 2 0 a.1 (0000) 1 (3) (3) (3) (3) 0 3 0 a.1 (0000) (3) (3, 3) (3, 3) (3, 3) (3, 3) 0 3 0 A.1 (1111) (3) (3, 3) (9) (9) (9) 0 4 0 a.1 (0000) (3, 3) (9, 3) (3, 3) (3, 3) (3, 3) 0 4 0 a.2 (1000) (3, 3) (9, 3) (3, 3) (3, 3) (3, 3) 0 4 0 a.3 (2000) (3, 3) (9, 3) (3, 3) (3, 3) (3, 3) 0 4 0 a.3∗ (2000) (3, 3) (3, 3, 3) (3, 3) (3, 3) (3, 3) 1 5 1 a.1 (0000) (9, 3) (9, 3) (3, 3) (3, 3) (3, 3) 0

For the order of the cosets s2, σ3,..., σm−1 we cannot ensure the upper bound 3, in general. However, in the cases m = 4, n = 5 and m = 5, n = 6, ρ = ±1 to be investigated, this is 3 3 possible: for m = 4, n = 5, we have s2 = σ3 = σ4 = 1, and for m = 5, n = 6, ρ = ±1 we have 3 3 3 1−ρ(β−1) −ρ −ρ σ3 = σ4 = σ5 = 1 and s2 = σ4 , where σ4 = σm−1 = τe+1 = τ4 ∈ γ2(M1). Consequently, we must only determine the order of the coset of the generator y with third power 3 y = σ3.

2.3.4. The 3-class group Cl3(N2) ' M2/γ2(M2).

M2 = hx, γ2(G)i = hx, s2, σ3, . . . , σm−1, τ3, . . . , τei,

γ2(M2) = hs3, σ4, . . . , σm−1i.

Since σ4, . . . , σm−1 ∈ γ2(M2) for m ≥ 5, the order of the cosets σ4,..., σm−1 equals 1. 3 3 The relation σ3σ4σ5 = 1 for third powers implies ord(σ3) ≤ 3. 3 −3 −3 −3ρ Since only the case e = 3 is to be investigated, we have τ3 = τ4 = τe+1 = σm−1 ∈ γ2(M2), because ρ = 0 for m = 4 and σm−1 ∈ γ2(M2) for m ≥ 5. 3 −ρβ −1 −ρ(β+1) Finally, we have s2 = σ4σm−1τ4 = σ4σm−1 ∈ γ2(M2), since σ4 ∈ γ2(M2) for m ≥ 4, ρ = 0 for m = 4, and σm−1 ∈ γ2(M2) for m ≥ 5, whence ord(s2) ≤ 3. Therefore, it only remains to investigate the order of the coset of the generator x with third power 3 x = τ3.

2.3.5. The 3-class groups Cl3(Ni) ' Mi/γ2(Mi), 3 ≤ i ≤ 4.

M3 = hxy, γ2(G)i = hxy, s2, σ3, . . . , σm−1, τ3, . . . , τei,

γ2(M3) = hs3t3, γ4(G)i, −1 −1 M4 = hxy , γ2(G)i = hxy , s2, σ3, . . . , σm−1, τ3, . . . , τei, −1 γ2(M4) = hs3t3 , γ4(G)i, γ4(G) = hσ4, . . . , σm−1, τ4, . . . , τei.

Since σ4, . . . , σm−1, τ4, . . . , τe ∈ γ4(G) < γ2(Mi) for e ≥ 4, m ≥ 5, the order of the cosets σ4,..., σm−1 and τ4,..., τe equals 1. 3 3 3 3 Due to the relations σ3σ4σ5 = 1 and τ3 τ4 τ5 = 1 for third powers, we have ord(σ3) ≤ 3 and ord(τ3) ≤ 3. 3 −ρβ −1 Finally, we have s2 = σ4σm−1τ4 ∈ γ2(Mi), since σ4, τ4 ∈ γ2(Mi) for m ≥ 4, e ≥ 3, ρ = 0 for m = 4 and σm−1 ∈ γ2(Mi) for m ≥ 5, whence ord(s2) ≤ 3. Thus, it remains only to determine the order of the coset of the generator xy resp. xy−1, for i = 3 resp. i = 4. PRINCIPALISATION ALGORITHM 9

Lemma 2.1. For each of the four 3-class groups Cl3(Ni) ' Mi/γ2(Mi) (1 ≤ i ≤ 4), the decision 3 if Cl3(Ni) is of type (9, 3) or of type(3, 3, 3) in the case of 3-class number h3(Ni) = 3 , depends exclusively on the order of the generator gi of Mi = hgi, γ2(G)i with respect to the commutator group γ2(Mi). The order of all the generators s2, σ3, . . . , σm−1, τ3, . . . , τe of γ2(G) with respect to γ2(Mi) is bounded from above by 3. After the preliminaries in the last three sections we come to the details of the rows of table 1 in the following four sections.

2.4. The smallest groups G of coclass cc(G) = 2 with bicyclic center, m = 4, n = 5. These groups satisfy the following special relations. n = 5 = 2m − 3, s = e = n − m + 2 = 3 = m − 1,

[χs(G), γe(G)] = [G, γ3(G)] = γ4(G) = 1,

σ4 = 1, τ4 = 1, 3 3 σ3 = 1, τ3 = 1, 3 s2 = 1,

γ2(G) = hs2i × hσ3i × hτ3i of type (3, 3, 3),

γ3(G) = hσ3i × hτ3i = ζ1(G) of type (3, 3). The commutator subgroups of the maximal normal subgroups are given by. −α 1−β γ2(M1) = ht3i = hσ3 τ3 i, γ−1 δ γ2(M2) = hs3i = hσ3 τ3 i, γ−α−1 δ−β+1 γ2(M3) = hs3t3i = hσ3 τ3 i, −1 α+γ−1 β+δ−1 γ2(M4) = hs3t3 i = hσ3 τ3 i. 3 −1 In table 3 we calculate the third powers gi of the generators g1 = y, g2 = x, g3 = xy, g4 = xy of the maximal normal subgroups Mi = hgi, γ2(G)i and the generators of the commutator groups γ2(Mi) with 1 ≤ i ≤ 4 for each of the seven isomorphism classes of groups G with m = 4, n = 5 [26, pp.1–3]. The principalisation types of these isomorphism classes are all diﬀerent. Generally, the third powers of g3, g4 are given by 3 α+γ β+δ −1 3 α−γ β−δ (xy) = σ3 τ3 and (xy ) = σ3 τ3 .

Table 3. Parameters, third powers, and generators for m = 4, n = 5, e = 3, k = 0

3 3 3 −1 3 −1 Type α β γ δ y x (xy) (xy ) t3 s3 s3t3 s3t3 −1 −1 −1 −1 b.10 0 0 0 0 σ3 τ3 1 1 τ3 σ3 σ3 τ3 σ3 τ3 −1 −1 −1 2 −1 c.21 0 0 0 1 σ3 τ3 τ3 τ3 τ3 σ3 τ3 σ3 τ3 σ3 −2 2 −1 −1 −1 −1 c.18 0 −1 0 1 σ3 τ3 1 τ3 τ3 σ3 τ3 σ3 σ3 τ3 −1 −1 −2 −2 2 −2 D.10 0 0 −1 1 σ3 τ3 σ3 τ3 σ3τ3 τ3 σ3 τ3 σ3 τ3 σ3 −1 −1 −1 2 −2 −2 2 −2 −2 G.19 0 −1 −1 0 σ3 τ3 σ3 τ3 σ3τ3 τ3 σ3 σ3 τ3 σ3 τ3 2 2 −1 −1 H.4 1 1 1 1 σ3 τ3 σ3τ3 1 σ3 τ3 σ3 τ3 σ3τ3 2 2 −1 −2 −1 D.5 1 1 −1 1 σ3 τ3 τ3 σ3 σ3 σ3 τ3 τ3 σ3 τ3

The order of the coset of gi ∈ Mi with respect to γ2(Mi) is bounded from above by 3, if and 3 only if the third power gi is contained in γ2(Mi). 10 DANIEL C. MAYER

Theorem 2.4. Let K be a base field with 3-class group Cl3(K) of type (3, 3). Suppose that the 2 n second 3-class group G = Gal(F3(K)|K) of K is of order |G| = 3 and of class cl(G) = m − 1, where m = 4 and n = 5, i.e., that G is one of the smallest groups of coclass cc(G) = 2 with invariant e = 3 and bicyclic center ζ1(G). 1 Then the structure of the 3-class groups of the first Hilbert 3-class field F3(K) of K and of the four unramified cyclic cubic extensions N1,...,N4 of K is given by table 4, in dependence on the principalisation type κ of K. The invariant ε denotes the number of elementary abelian 3-class groups Cl3(Ni) for each principalisation type.

Table 4. Elementary abelian 3-class groups for m = 4, n = 5, e = 3, k = 0

1 Type κ Cl3(F3(K)) Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) ε b.10 (0043) (3, 3, 3) (9, 3) (9, 3) (3, 3, 3) (3, 3, 3) 2 c.21 (0231) (3, 3, 3) (9, 3) (9, 3) (9, 3) (9, 3) 0 c.18 (0313) (3, 3, 3) (9, 3) (9, 3) (3, 3, 3) (9, 3) 1 D.10 (2241) (3, 3, 3) (9, 3) (9, 3) (3, 3, 3) (9, 3) 1 G.19 (2143) (3, 3, 3) (9, 3) (9, 3) (9, 3) (9, 3) 0 H.4 (4443) (3, 3, 3) (3, 3, 3) (3, 3, 3) (9, 3) (3, 3, 3) 3 D.5 (4224) (3, 3, 3) (3, 3, 3) (9, 3) (3, 3, 3) (9, 3) 2

1 Proof. The structure of the 3-class group Cl3(F3(K)) of the ﬁrst Hilbert 3-class ﬁeld of K can be obtained from the parameters m = 4 and e = 3 by means of the isomorphism [1]

1 2 1 Cl3(F3(K)) ' Gal(F3(K)|F3(K)) ' γ2(G) ' A(3, m − 2) × A(3, e − 2) = A(3, 2) × A(3, 1) 1 2 of Cl3(F3(K)) with the commutator group γ2(G) of G = Gal(F3(K)|K). The structure of the 3-class groups Cl3(Ni) is a consequence of table 3, since we have the isomorphism [1]

1 2 2 1 Cl3(Ni) ' Gal(F3(Ni)|Ni) ' Gal(F3(K)|Ni)/Gal(F3(K)|F3(Ni)) ' Mi/γ2(Mi), taking into consideration the preliminaries in the sections 2.3.3, 2.3.4, and 2.3.5. Here we use the equivalence of the following statements.

3 3 Cl3(N1) ' A(3, 3) ⇐⇒ g1 = y 6∈ ht3i = γ2(M1), 3 3 Cl3(N2) ' A(3, 3) ⇐⇒ g2 = x 6∈ hs3i = γ2(M2), 3 3 Cl3(N3) ' A(3, 3) ⇐⇒ g3 = (xy) 6∈ hs3t3i = γ2(M3), 3 −1 3 −1 Cl3(N4) ' A(3, 3) ⇐⇒ g4 = (xy ) 6∈ hs3t3 i = γ2(M4). Corollary 2.4.1. For each of these seven isomorphism classes of the second 3-class group G, a 3-class group Cl3(Ni) with 1 ≤ i ≤ 4 is elementary abelian, if and only if the norm class group

Proof. This is an immediate consequence of the principalisation types κ of these seven isomorphism classes. PRINCIPALISATION ALGORITHM 11

Corollary 2.4.2. If K is a quadratic base ﬁeld, then the three total principalisation types b.10, c.21, c.18 are impossible, due to class number relations, and the remaining four partial principalisation types D.10, G.19, H.4, D.5 are characterised uniquely by the invariant ε.

Proof. For the principalisation types b.10, c.21, c.18 with singulet κ(1) = 0, the entire 3-class group Cl3(K) becomes principal in N1. Hence, a quadratic base field K must be real. The unramified cyclic cubic extension N1 is an S3-field of type α with even 3-exponent of the 3-class number h3(N1) in contradiction to Cl3(N1) ' A(3, 3) [22, Prop.5.3 and 5.4]. 2 Example 2.1. The first occurrences of second 3-class groups G = Gal(F3(K)|K) of coclass cc(G) = 2 with invariants√ m = 4, n = 5, e = 3 and bicyclic center among the quadratic base fields K = Q( D) with discriminant −106 < D < 0 resp. 0 < D < 107 and with 3-class group of type (3, 3) turned out to be the following. • The smallest value |D| of the discriminant of a complex quadratic field K with principalisation type D.10 resp. D.5 is 4 027 [29] resp. 12 131 [16]. • The smallest discriminant D of a real quadratic field K with principalisation type D.10 resp. D.5 is 422 573 resp. 631 769. Conjecture 2.1. The principalisation types G.19 and H.4 cannot occur with invariants m = 4, n = 5, and k = 0 for quadratic base fields, since the corresponding second 3-class groups G have terminal successors of the same principalisation type with invariants m = 5, n = 6, and k = 1. (Weak or restricted leaf conjecture.) 2.5. The smallest groups G of coclass cc(G) = 2 with cyclic center, m = 5, n = 6. These groups satisfy the following special relations. n = 6 = 2m − 4, s = 4 = m − 1, e = n − m + 2 = 3 = m − 2,

[χs(G), γe(G)] = [G, γ3(G)] = γ4(G) = γm−1(G) > 1,

σ5 = 1, τ5 = 1, 3 3 σ4 = 1, τ4 = 1, 3 −3 3 −3 σ3 = σ4 = 1, τ3 = τ4 = 1, 3 1−ρ(β−1) s2 = σ4 , 3 s2 = 1 ⇐⇒ β = 0, ρ = −1 or β = −1, ρ = 1, −ρ τ4 = σ4 , ( 3 (9, 3, 3), if s2 6= 1, γ2(G) = hs2, σ3, σ4, τ3i of type 3 (3, 3, 3, 3), if s2 = 1,

γ3(G) = hσ3, σ4, τ3i of type (3, 3, 3),

γ4(G) = hσ4i = ζ1(G) of type (3). The commutator subgroups of the maximal normal subgroups are given by. The dependencies on the parameters α, γ disappear, since they occur in the exponent of σ4 ∈ γ4(G), but each γ2(Mi) with 1 ≤ i ≤ 4 contains γ4(G) = hσ4i = hτ4i.

−ρδ ρ−α 1−β −ρδ 1−β γ2(M1) = ht3, σ4i = hσ3 σ4 τ3 , σ4i = hσ3 τ3 , σ4i, ρβ−1 γ−1 δ ρβ−1 δ γ2(M2) = hs3, σ4i = hσ3 σ4 τ3 , σ4i = hσ3 τ3 , σ4i, ρ(β−δ)−1 δ−β+1 γ2(M3) = hs3t3, σ4i = hσ3 τ3 , σ4i, −1 ρ(β+δ)−1 β+δ−1 γ2(M4) = hs3t3 , σ4i = hσ3 τ3 , σ4i. 3 −1 In table 5 we calculate the third powers gi of the generators g1 = y, g2 = x, g3 = xy, g4 = xy of the maximal normalsubgroups Mi = hgi, γ2(G)i and the generators of the commutator groups 12 DANIEL C. MAYER

Table 5. Parameters, third powers, and generators for m = 5, n = 6, e = 3, k = 1

3 3 3 −1 3 −1 Type isom. cl. β δ ρ y x (xy) (xy ) t3 s3 s3t3 s3t3 −1 −1 −1 −1 b.10 3 0 0 1 σ3 τ3 1 1 τ3 σ3 σ3 τ3 σ3 τ3 2 2 −1 −1 H.4 4 1 1 1 σ3 τ3 σ3τ3 1 σ3 τ3 σ3 τ3 σ3τ3 −1 −1 −1 2 −2 −2 2 −2 −2 G.19 2 −1 0 1 σ3 τ3 σ3 τ3 σ3τ3 τ3 σ3 σ3 τ3 σ3 τ3 −1 −1 −1 −1 b.10 3 0 0 −1 σ3 τ3 1 1 τ3 σ3 σ3 τ3 σ3 τ3

γ2(Mi) (1 ≤ i ≤ 4) modulo σ4 for each of the twelve isomorphism classes of groups G with m = 5, n = 6, ρ = ±1 [26, pp.4–7]. Several of these classes have the same pricipalisation type and the same parameters β, δ, ρ. Generally, the third powers of g3, g4 modulo σ4 are given by 3 ρ(β+δ) α+γ+ρ(β+δ) β+δ ρ(β+δ) β+δ −1 3 ρ(δ−β) α−γ+ρβ β−δ ρ(δ−β) β−δ (xy) = σ3 σ4 τ3 ≡ σ3 τ3 and (xy ) = σ3 σ4 τ3 ≡ σ3 τ3 . The order of the coset of gi ∈ Mi with respect to γ2(Mi) is bounded from above by 3, if and 3 only if the third power gi is contained in γ2(Mi).

Theorem 2.5. Let K be a base field with 3-class group Cl3(K) of type (3, 3). Suppose that the 2 n second 3-class group G = Gal(F3(K)|K) is of order |G| = 3 and class cl(G) = m − 1, where n = 6 and m = 5, such that [χs(G), γe(G)] = γm−1(G), k = 1, i.e., that G is one of the smallest groups of coclass cc(G) = 2 with invariant e = 3 and cyclic center ζ1(G). 1 Then the structure of the 3-class groups of the first Hilbert 3-class field F3(K) of K and of the four unramified cyclic cubic extensions N1,...,N4 of K is given by table 6, in dependence on the principalisation type κ of K and on the relational exponents β, ρ of G. The invariant ε denotes the number of elementary abelian 3-class groups Cl3(Ni) for each principalisation type.

Table 6. Elementary abelian 3-class groups for m = 5, n = 6, e = 3, k = 1

1 Type κ β ρ Cl3(F3(K)) Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) ε b.10 (0043) 0 1 (9, 3, 3) (9, 3) (9, 3) (3, 3, 3) (3, 3, 3) 2 H.4 (4443) 1 1 (9, 3, 3) (3, 3, 3) (3, 3, 3) (9, 3) (3, 3, 3) 3 G.19 (2143) −1 1 (3, 3, 3, 3) (9, 3) (9, 3) (9, 3) (9, 3) 0 b.10 (0043) 0 −1 (3, 3, 3, 3) (9, 3) (9, 3) (3, 3, 3) (3, 3, 3) 2

1 Proof. Similarly as in the proof of theorem 2.4, the structure of the 3-class group Cl3(F3(K)) of the ﬁrst Hilbert 3-class ﬁeld of K is a consequence of m = 5, e = 3, and the isomorphism ( 1 A(3, m − 2) × A(3, e − 2) = A(3, 3) × A(3, 1) falls ρ 6≡ β − 1 (mod 3), Cl3(F (K)) ' γ2(G) ' 3 A(3, m − 3) × A(3, e − 1) = A(3, 2) × A(3, 2) falls ρ ≡ β − 1 (mod 3).

The structure of the 3-class groups Cl3(Ni) ' Mi/γ2(Mi) follows from table 5, if we take into consideration the preparations in the sections 2.3.3, 2.3.4, and 2.3.5. Corollary 2.5.1. For each of these twelve isomorphism class of the second 3-class group G, a 3-class group Cl3(Ni) with 1 ≤ i ≤ 4 is elementary abelian, if and only if the norm class group

NormNi|K (Cl3(Ni)) becomes principal either in none or in three of the extensions Nj (1 ≤ j ≤ 4). The extensions with elementary abelian 3-class group always satisfy the condition (B) of Taussky [31], i.e., they have a principalisation without ﬁxed point. Proof. This is an immediate consequence of the principalisation types κ of these twelve isomorphism classes. PRINCIPALISATION ALGORITHM 13

Corollary 2.5.2. If K is a quadratic base ﬁeld, then the total principalisation type b.10 is impossible, due to class number relations, and the remaining two partial principalisation types H.4 and G.19 are uniquely characterised by the invariant ε.

Proof. For the principalisation type b.10 with singulet κ(1) = 0, the entire 3-class group Cl3(K) becomes principal in N1. In the case of a quadratic base field K this yields a similar contradiction as in the proof of corollary 2.4.2. 2 Example 2.2. The first occurrences of second 3-class groups G = Gal(F3(K)|K) of coclass cc(G) = 2 with invariants√ m = 5, n = 6, e = 3, ρ = ±1 and cyclic center among the quadratic base fields K = Q( D) with discriminant −106 < D < 0 resp. 0 < D < 107 and with 3-class group of type (3, 3) turned out to be the following. • The smallest value |D| of the discriminant of a complex quadratic field K with principalisation type H.4 resp. G.19 is 3 896 [16] resp. 12 067 [16]. • The smallest discriminant D of a real quadratic field K with principalisation type G.19 resp. H.4 is 214 712 resp. 957 013. 2.6. The other groups of coclass cc(G) = 2, m ≥ 5, n ≥ 6. These groups satisfy the following general relations. For m ≥ 6, we have σm−2, σm−1 ∈ γ4(G). For m ≥ 5, the case ρ = ±1 has been investigated in the preceding section already, and we only have to consider the remaining possibility ρ = 0. Since σm−1 ∈ γ4(G) also for m = 5, the following congruences modulo γ4(G) are valid generally. The appearing dependencies on α, γ, ρ vanish.

3 ρ(β+δ) α+γ+ρ(β+δ) β+δ β+δ (xy) = σm−2 σm−1 τ3 ≡ τ3 (mod γ4(G)), −1 3 ρ(δ−β) α−γ+ρβ β−δ β−δ (xy ) = σm−2 σm−1 τ3 ≡ τ3 (mod γ4(G)), −β −ρδ −α 1−β t3 = τ3τ4τ3 σm−2σm−1 ≡ τ3 (mod γ4(G)), −1 −1 ρβ γ δ −1 δ s3 = σ3 σ4 σm−2σm−1τ3 ≡ σ3 τ3 (mod γ4(G)), −1 δ−β+1 s3t3 ≡ σ3 τ3 (mod γ4(G)), −1 −1 β+δ−1 s3t3 ≡ σ3 τ3 (mod γ4(G)). 3 In the following table we calculate the third powers gi of the generators g1 = y, g2 = x, g3 = −1 xy, g4 = xy of the maximal normal subgroups Mi = hgi, γ2(G)i and the generators of the commutator groups γ2(Mi) with 1 ≤ i ≤ 4 modulo γ4(G) for each of the isomorphism classes of groups G with m ≥ 5, n ≥ 6, e = 3. Several of these classes have the same principalisation type and the same parameters β, δ. The number of isomorphism classes in brackets concerns the single case m = 5 [26, pp.8–9], the number in parentheses odd values of m ≥ 7 [26, pp.8–9, pp.16–19] and the number between even values of m ≥ 6 [26, pp.10–12, pp.13–15]. The order of the coset of gi ∈ Mi with respect to γ2(Mi) is bounded from above by 3, if and 3 only if the third power gi is contained in γ2(Mi).

Theorem 2.6. Let K be a base field with 3-class group Cl3(K) of type (3, 3). Suppose that the 2 n second 3-class group G = Gal(F3(K)|K) of K is of order |G| = 3 and of class cl(G) = m − 1, where n ≥ 6 and m = n − 1, i.e., that G is of coclass cc(G) = 2 with invariant e = 3. In the case m = 5, n = 6 let [χs(G), γe(G)] = 1. Then the structure of the 3-class groups Cl3(Ni) of the four unramified cyclic cubic extensions Ni of K is given by table 8, in dependence on the principalisation type κ of K. The invariant ε denotes the number of elementary abelian 3-class groups Cl3(Ni) for each principalisation type. Generally, the first two 3-class groups Cl3(N1) and Cl3(N2) are nearly homocyclic.

Proof. The structure of the ﬁrst 3-class group Cl3(N1) is given here only for the sake of complete- ness and is contained in the statement of theorem 2.2 already. Similarly as in the proof of theorem 2.4, ergibt sich the structure of the other 3-class groups 14 DANIEL C. MAYER

Table 7. Parameters, third powers, and generators for m ≥ 5, n ≥ 6, e = 3

3 3 3 −1 3 −1 Type isom. cl. β δ y x (xy) (xy ) t3 s3 s3t3 s3t3 −1 −1 −1 −1 b.10 1 7 9 0 0 σ3 τ3 1 1 τ3 σ3 σ3 τ3 σ3 τ3 −1 −1 −1 −1 d.19 1 2 1 0 0 σ3 τ3 1 1 τ3 σ3 σ3 τ3 σ3 τ3 −1 −1 −1 −1 d.23 1 1 1 0 0 σ3 τ3 1 1 τ3 σ3 σ3 τ3 σ3 τ3 −1 −1 −1 −1 d.25 1 2 1 0 0 σ3 τ3 1 1 τ3 σ3 σ3 τ3 σ3 τ3 −1 −1 −1 2 −1 c.21 1 1 1 0 1 σ3 τ3 τ3 τ3 τ3 σ3 τ3 σ3 τ3 σ3 −1 −1 −1 2 −1 E.8 1 1 1 0 1 σ3 τ3 τ3 τ3 τ3 σ3 τ3 σ3 τ3 σ3 −1 −1 −1 2 −1 E.9 1 2 1 0 1 σ3 τ3 τ3 τ3 τ3 σ3 τ3 σ3 τ3 σ3 −1 −1 −1 2 −1 G.16 1 8 9 0 1 σ3 τ3 τ3 τ3 τ3 σ3 τ3 σ3 τ3 σ3 −2 2 −1 −1 −1 −1 c.18 1 1 1 −1 1 σ3 τ3 1 τ3 τ3 σ3 τ3 σ3 σ3 τ3 −2 2 −1 −1 −1 −1 E.6 1 1 1 −1 1 σ3 τ3 1 τ3 τ3 σ3 τ3 σ3 σ3 τ3 −2 2 −1 −1 −1 −1 E.14 1 2 1 −1 1 σ3 τ3 1 τ3 τ3 σ3 τ3 σ3 σ3 τ3 −2 2 −1 −1 −1 −1 H.4 1 8 9 −1 1 σ3 τ3 1 τ3 τ3 σ3 τ3 σ3 σ3 τ3

Table 8. Elementary abelian 3-class groups for m ≥ 5, n ≥ 6, e = 3

Type κ Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) ε b.10 (0043) A(3, m − 1) or A(3, m − 2) (9, 3) (3, 3, 3) (3, 3, 3) 2 d.19 (4043) A(3, m − 1) (9, 3) (3, 3, 3) (3, 3, 3) 2 d.23 (1043) A(3, m − 1) (9, 3) (3, 3, 3) (3, 3, 3) 2 d.25 (2043) A(3, m − 1) (9, 3) (3, 3, 3) (3, 3, 3) 2 c.21 (0231) A(3, m − 1) (9, 3) (9, 3) (9, 3) 0 E.8 (1231) A(3, m − 1) (9, 3) (9, 3) (9, 3) 0 E.9 (2231) A(3, m − 1) (9, 3) (9, 3) (9, 3) 0 G.16 (4231) A(3, m − 1) or A(3, m − 2) (9, 3) (9, 3) (9, 3) 0 c.18 (0313) A(3, m − 1) (9, 3) (3, 3, 3) (9, 3) 1 E.6 (1313) A(3, m − 1) (9, 3) (3, 3, 3) (9, 3) 1 E.14 (2313) A(3, m − 1) (9, 3) (3, 3, 3) (9, 3) 1 H.4 (3313) A(3, m − 1) or A(3, m − 2) (9, 3) (3, 3, 3) (9, 3) 1

Cl3(Ni) ' Mi/γ2(Mi) with 2 ≤ i ≤ 4 is a consequence of table 7, when the preparations in the sections 2.3.4 and 2.3.5 are taken into consideration. Corollary 2.6.1. As before, for these isomorphism classes of the second 3-class group G, the extensions with elementary abelian 3-class group satisfy the condition (B) of Taussky [31], i.e., they have a principalisation without ﬁxed point. However, here only the following weaker statement without admissible inversion is true: if a 3-class group Cl3(Ni) with 3 ≤ i ≤ 4 is elementary abelian, then the norm class group NormNi|K (Cl3(Ni)) becomes principal in either two or three of PRINCIPALISATION ALGORITHM 15 the extensions Nj (1 ≤ j ≤ 4).

Proof. This follows by evaluating the principalisation type κ of the isomorphism classes. Corollary 2.6.2. If K is a quadratic base field, then the four total principalisation types b.10, d.19, d.23, and d.25 are impossible, due to class number relations. The remaining eight principalisation types can only be partially characterised by the value of the invariant ε. For the isolation of the total principalisation types c.21 and c.18, the parity of the 3-exponent of the first 3-class number h3(N1) must be used, and for the separation of the partial principalisation types E.8/9 1 and G.16, resp. E.6/14 and H.4, the structure of the 3-classgroup Cl3(F3(K)) of the first Hilbert 3-class field of K is responsible.

Proof. For the principalisation types b.10, d.19, d.23, and d.25 with κ(2) = 0, the entire 3-class group Cl3(K) becomes principal in N2. Therefore a quadratic base field K must be real. The unramified cyclic cubic extension N2 is an S3-field of type α [22] with even 3-exponent of the 3-class number h3(N2) in contradiction to Cl3(N2) ' A(3, 3). µ The parity of the 3-exponent of the first 3-class number h3(N1) = 3 turns out to be even, µ = m − 1, for the total principalisation types c.21 and c.18 with κ(1) = 0, to be odd, µ = m − 1, for the partial principalisation types E.8/9 and E.6/14 with k = 0, and to be odd, µ = m − 2, for the partial principalisation types G.16 and H.4 with k = 1.

2 Example 2.3. The first occurrences of second 3-class groups G = Gal(F3(K)|K) of coclass cc(G) = 2 with invariants m ≥ 5, n √≥ 6, e = 3 and bicyclic center, k = 0, resp. cyclic center, k = 1, over the quadratic base fields Q( D) with 3-class group of type (3, 3) and discriminant −106 < D < 0 resp. 0 < D < 107 are summarised in table 9. Here |D| denotes the smallest absolute value of the discriminant of a complex quadratic field K and D the smallest discriminant of a real quadratic field K of the mentioned principalisation type. The cited earlier computations by Scholz and Taussky [29], by Heider and Schmithals [16], and by Brink [9] are confirmed.

Table 9. Examples for m ≥ 5, n ≥ 6, e = 3

Type κ m n k |D| ref. D c.21 (0231) 5 6 0 impossible 540 365 c.21 (0231) 7 8 0 impossible 1 001 957 E.8 (1231) 6 7 0 34 867 6 098 360 E.9 (2231) 6 7 0 9 748 [29] 342 664 E.9 (2231) 8 9 0 297 079 unknown G.16 (4231) 7 8 1 17 131 [16] 8 711 453 c.18 (0313) 5 6 0 impossible 534 824 E.6 (1313) 6 7 0 15 544 [16] 5 264 069 E.6 (1313) 8 9 0 268 040 unknown E.14 (2313) 6 7 0 16 627 [16] 3 918 837 E.14 (2313) 8 9 0 262 744 unknown H.4 (3313) 7 8 1 21 668 [9] 1 162 949 H.4 (3313) 9 10 1 446 788 unknown 16 DANIEL C. MAYER

Whereas in example 2.1 the parameters m = 4, n = 5 for the principalisation types D.10, D.5 and in example 2.2 the parameters m = 5, n = 6 for the ground variants of principalisation types H.4, G.19 are determined uniquly, we now have infinite families of second 3-class groups 2 G = Gal(F3(K)|K), for which the nilpotency class cl(G) + 1 = m for the principalisation types c.21, c.18 can take all odd values m ≥ 5, for the principalisation types E.8, E.9, E.6, E.14 all even values m ≥ 6, and for principalisation types H.4, G.16 all odd values m ≥ 7. A peculiarity of the principalisation types c.21 and c.18 is that their second 3-class groups G = 2 Gal(F3(K)|K) can occur exclusively as internal nodes on infinite chains. Concrete numerical realisations are known for the ground state with minimal nilpotency class m for each of these infinite families and for higher values of m for the principalisation types E.9, E.6, E.14 and H.4 of complex quadratic base fields and the principalisation type c.21 of real quadratic base fields. 2.7. The groups of coclass cc(G) ≥ 3, m ≥ 5, n ≥ 7. These groups satisfy the following general relations. 3 ρ(β+δ) α+γ+ρ(β+δ) β+δ (xy) = σm−2 σm−1 τe , −1 3 ρ(δ−β) α−γ+ρβ β−δ (xy ) = σm−2 σm−1 τe . −1 Therefore the order of the coset of xy ∈ M3 with respect to γ2(M3) and of the coset of xy ∈ M4 with respect to γ2(M4) is certainly bounded from above by 3, if m ≥ 6 and e ≥ 4, and thus σm−2, σm−1, τe ∈ γ4(G) < γ2(Mi) for i = 3, 4. It remains to investigate the fifteen isomorphism classes of groups with m = 5, n = 7, e = 4, for which [χs(G), γe(G)] = 1 and thus ρ = 0 [26, pp.34–35]. For these classes, third powers of the generators are given by

3 α+γ β+δ (xy) = σ4 τ4 , −1 3 α−γ β−δ (xy ) = σ4 τ4 , and therefore ord(xy) ≤ 3 and ord(xy−1) ≤ 3, independently from all parameters α, β, γ, δ, since σ4, τ4 ∈ γ4(G) < γ2(Mi) for i = 3, 4.

Theorem 2.7. Let K be a base field with 3-classgroup Cl3(K) of type (3, 3) and with second 3-class 2 n group G = Gal(F3(K)|K) of order |G| = 3 where n ≥ 7 and of coclass cc(G) ≥ 3, m ≤ n − 2, e ≥ 4. Then the structure of the 3-class groups of the four unramified cyclic cubic extensions Ni of K is almost homogeneous for Cl3(N1) and Cl3(N2) and elementary abelian of type (3, 3, 3) for Cl3(N3) and Cl3(N4), independently from the principalisation type κ of K. The number of elementary abelian 3-class groups Cl3(Ni) is always given by ε = 2. The extensions with elementary abelian 3-class group satisfy Taussky’s condition (B) [31], that is, a principalisation without fixed point.

Proof. The nearly homocyclic structure of the ﬁrst and second 3-class group, Cl3(N1), Cl3(N2), is contained in the statement of theorem 2.2 already. Similarly as in the proof of theorem 2.4, the elementary abelian structure of the third and fourth 3-class group, Cl3(Ni) ' Mi/γ2(Mi) with 3 ≤ i ≤ 4, is a consequence of the considerations at the beginning of this section, if we take into account the preparation in section 2.3.5. PRINCIPALISATION ALGORITHM 17

2 Example 2.4. The first occurrences of second 3-class groups G = Gal(F3(K)|K) of coclass cc(G) ≥ 3 with invariants m ≥ 5, n ≥ 7, e ≥ 4,√ and bicyclic center, k = 0, resp. cyclic center, k = 1, over the quadratic base fields K = Q( D) with 3-class group of type (3, 3) and discriminant −106 < D < 0 resp. 0 < D < 107 are summarised in table 10. Here |D| denotes the smallest absolute value of the discriminant of a complex quadratic field K and D the smallest discriminant of a real quadratic field K of the mentioned principalisation type. The cited earlier computations by Brink [9] are confirmed.

Table 10. Examples for m ≥ 5, n ≥ 6, e = 3

1 Type κ m n e k Cl3(F3(K)) |D| ref. D b.10 (0043) 6 8 4 1 impossible 710 652 d.19 (4043) 6 8 4 0 impossible 2 328 721 d.23 (1043) 6 8 4 0 impossible 1 535 117 d.25∗ (0143) 7 10 5 0 impossible 8 491 713 F.7 (3443) 6 9 5 0 124 363 unknown F.11 (1143) 6 9 5 0 27 156 [9] unknown F.12 (1343) 6 9 5 0 31 908 [9] unknown F.12 (1343) 8 11 5 0 249 371 unknown F.13 (3143) 6 9 5 0 67 480 [9] 8 321 505 F.13 (3143) 8 11 5 0 159 208 8 127 208 G.16 (1243) 7 10 5 1 (27, 9, 9, 3) 290 703 unknown G.16 (1243) 7 10 5 1 (9, 9, 9, 9) 135 059 unknown G.19 (2143) 7 10 5 1 (27, 9, 9, 3) 96 827 unknown G.19 (2143) 7 10 5 1 (9, 9, 9, 9) 199 735 unknown H.4 (3343) 7 10 5 1 (27, 9, 9, 3) 256 935 unknown H.4 (3343) 7 10 5 1 (9, 9, 9, 9) 186 483 unknown

2 Similarly as in example 2.3, these are infinite families of second 3-class groups G = Gal(F3(K)|K), but additionally to the nilpotency class cl(G) + 1 = m the invariant e = n − m + 2 can also take infinitely many values. For the principalisation type b.10 all even values m ≥ 6 and all even values 4 ≤ e ≤ m − 2 are possible, for the types F.7, F.11, F.12, and F.13 all even values m ≥ 6 and all odd values 5 ≤ e ≤ m − 1 can be assumed, and for the types G.16, G.19, and H.4 all odd values m ≥ 7 and all odd values 5 ≤ e ≤ m − 2 are admissible. The very rare principalisation types d.19, d.23, and d.25 play a unique exceptional role [23, Th.3.4], 2 since their second 3-class groups G = Gal(F3(K)|K) can be either terminal nodes (leaves) with even values m ≥ 6 and even values 4 ≤ e ≤ m − 2 or internal nodes on infinite chains with odd values m ≥ 7 and odd values 5 ≤ e ≤ m − 2. Concrete numerical realisations are known for the ground state with minimal nilpotency class m for each of these infinite families of complex quadratic base fields, but only for certain families of real quadratic base fields. 18 DANIEL C. MAYER

3. Implementation of the principalisation algorithm for quadratic fields In this section we describe the new principalisation algorithm and its implementation with the aid of program scripts for the number theoretical computer algebra system PARI/GP [27]. We refer to the utilized methods of this software package, which are printed in typewriter font.

3.1. Generating polynomials for non-Galois cubic ﬁelds L.

3.1.1. Simply real cubic fields. Generating polynomials p(X) = X3 − bX2 + cX − d of third degree for simply real cubic fields L of signature (1, 1) are obtained in the following manner. Suppose the intended upper bound for the absolute value of the field discriminant is |d(L)|√ ≤ U. Then candidates for the coefficients b, c, and d run in three nested loops 1 ≤ b ≤ b3 + 2 4 Uc, √ √ b2+ U b2−3+2 U 1 ≤ c ≤ b 3 c, and 1 ≤ d ≤ b 6 c, with bounds due to Godwin and Angell, given by Fung and Williams [13]. For each triplet (b, c, d) the following tests are performed. (1) Reducible polynomials p(X) are eliminated with the aid of polisirreducible(). (2) For irreducible polynomials the discriminant d(L) of the generated cubic field L is calculated by means of nfdisc() and tested for −U ≤ d(L) < 0. (3) By poldisc() the discriminant d(p) of the polynomial is computed and its index i(p) is determined, using the formula d(p) = i(p)2 · d(L). Polynomials with indices bigger than the bound p √ 124b2 + 432b + 4 U + 729 √ 3 3 are skipped, thus discouraging superfluous isomorphic fields. (4) Field discriminants d(L) are restricted to fundamental discriminants by isfundamental(). (5) Finally, the 3-class group Cl3(K) of the complex quadratic subfield K of the Galois closure N of L, which is unramified over K with discriminant d(N) = d(K)3, is restricted to the type (3, 3) with the aid of quadclassunit(), thereby eliminating the numerous cyclic 3-class groups.

3.1.2. Totally real cubic fields. Trace free generating polynomials p(X) = X3 − cX − d for totally real cubic fields L of signature (3, 0) are collected in the following way. If the desired upper bound for the field discriminant is d(L) ≤ U, then candidates for the coefficients c and d run in two √ q 4c3 nested loops 1 ≤ c ≤ b Uc and 1 ≤ d ≤ b 27 c, with bounds given by Llorente and Quer [19]. For each pair (c, d) the following tests are performed. (1) Reducible polynomials p(X) are skipped with the aid of polisirreducible(). (2) For irreducible polynomials the discriminant d(L) of the generated cubic field L is calculated by means of nfdisc() and checked for 0 < d(L) ≤ U. (3) A further bound is imposed on the linear coefficient √ ( d(L) b c, if 27 | d(L), c ≤ 3 bpd(L)c, otherwise.

(4) By poldisc() the discriminant d(p) of the polynomial is computed and its index i(p) is determined, using the formula d(p) = i(p)2 · d(L). Polynomials with indices bigger than the bound ( b2p c c, if 27 | d(L), √ 3 b2 cc, otherwise, are eliminated. (5) Field discriminants d(L) are restricted to fundamental discriminants by isfundamental(). (6) Finally, the 3-class group Cl3(K) of the real quadratic subﬁeld K of the Galois closure N of L is restricted to the type (3, 3) with the aid of quadclassunit(). PRINCIPALISATION ALGORITHM 19

3.2. Structure of 3-class groups Cl3(N) of S3-fields N. The generating polynomials of section 3.1 are stored as quadruplets (D, b, c, d) for D < 0, ordered by descending discriminants D = d(K) = d(L), resp. as triplets (D, c, d) for D > 0, ordered by ascending discriminants D = d(K) = d(L). The bounds of [13] resp. [19] provide a warranty that, for each discriminant D, generating polynomials for all four non-isomorphic cubic fields L sharing the same discriminant d(L) = D are contained in the list. Now the polynomials are iterated through the list in a single loop and for each of them the following steps are executed. (1) The regulator R(L) and the class number h(L) of L are calculated with the aid of bnfinit() using the flag 1, which ensures that a system of fundamental units of L is determined. An indicator is stored, if the class number h(L) is divisible by 9 or 27. (2) By means of polcompositum(), applied to the cubic polynomial p(X) = X3−bX2+cX−d, resp. p(X) = X3 − cX − d, and the quadratic polynomial q(X) = X2 − D, a generating polynomial of sixth degree for the Galois closure N of L is calculated. (3) The polynomial of sixth degree is used to determine the structure of the class group Cl(N) of the normal field N with the aid of bnfinit(), where the flag is set to 1. An indicator is stored if the first three invariants (n1, n2, n3,...) of the class group structure are all divisible by 3, that is, if the 3-class group Cl3(N) of N is elementary abelian of type (3, 3, 3). 3 For the quadruplets N1,...,N4 of S3-fields, which share the same discriminant D , the structure of their 3-class groups Cl(Ni) (1 ≤ i ≤ 4) determines the number ε of elementary abelian groups of type (3, 3, 3), the principalisation type κ(K), and the structure of the second 3-class group 2 G = Gal(F3(K)|K) of the quadratic field K by the Theorems 2.3, 2.4, and 2.5. If some 3-class group Cl(Ni) is of type (3, 3), then G is of coclass cc(G) = 1 and we can terminate the algorithm. Otherwise G is of coclass cc(G) ≥ 2 and the algorithm stops, provided that no indicator for 9 | h(Li) has been set for any 1 ≤ i ≤ 4.

1 3.3. First Hilbert 3-class field F3(K) of K. For the quadruplets L1,...,L4 of cubic fields, which share the same discriminant D and have been marked by an indicator for 9 | h(Li) for some 2 1 ≤ i ≤ 4, we must append a further step, if the second 3-class group G = Gal(F3(K)|K) of the quadratic field K is of coclass cc(G) ≥ 2. A generating polynomial of eighteenth degree for 1 the first Hilbert 3-class field F3(K) = Ni · Lj of K is calculated by means of polcompositum(), applied to the generating polynomial of sixth degree of Ni and the cubic generating polynomial of Lj for some 1 ≤ i 6= j ≤ 4. The polynomial of eighteenth degree is used to determine the structure 1 1 of the class group Cl(F3(K)) of the first Hilbert 3-class field F3(K) of the quadratic field K with the aid of bnfinit(), where the flag is set to 1. Whereas computations for fields of third and sixth degree are a matter of less than a second, the CPU time for a field of degree 18 can reach a few minutes occasionally. Together with the structure of the 3-class groups Cl(Ni) (1 ≤ i ≤ 4) and the number ε of ele- 1 mentary abelian groups of type (3, 3, 3), the structure of the 3-class group Cl3(F3(K)) determines 2 the principalisation type κ(K) and the structure of the second 3-class group G = Gal(F3(K)|K) of the quadratic field K by means of the theorems [22, Th.5.1,5.2,5.3] and the Theorems 2.6 and 2.7,

4. Numerical results for 4 596 quadratic fields By means of the PARI/GP programs in section 3, the principalisation type κ and the structure 2 of the second 3-class√ group G = Gal(F3(K)|K) has been determined for the 2 020 complex qua- 6 dratic fields√ K = Q( D) with discriminant −10 < D < 0 and for the 2 576 real quadratic fields 7 K = Q( D) with discriminant 0 < D < 10 and 3-class group Cl3(K) of type (3, 3). The results of these extensive computations reveal reliable statistical tendencies for a total of 4 596 quadratic base fields. Similarly as in [23] we define an isomorphism invariant ν = ν(K) for each of these quadratic fields K, which characterizes the amount of total principalisations of K. 20 DANIEL C. MAYER

Definition 4.1. Let 0 ≤ ν ≤ 4 be the number ν = #{1 ≤ i ≤ 4 | κ(i) = 0} of unramified cyclic cubic extension fields Ni of K, in which the entire 3-class group Cl3(K) of K becomes principal.

Table 11. Principalisation types with invariant 3 ≤ ν ≤ 4 for D > 0

1 type κ h3(L1) Cl3(N1) ε Cl3(F3(K)) min. D ref. freq. a.1 (0000) 9 (9, 9) 0 (9, 9) 62 501 [16] 147 a.2 (1000) 3 (9, 3) 0 (3, 3) 72 329 [16] o 1 386 a.3 (2000) 3 (9, 3) 0 (3, 3) 32 009 [16] a.3* (2000) 3 (3, 3, 3) 1 (3, 3) 142 097 [21] 697 a.1↑ (0000) 27 (27, 27) 0 (27, 27) 2 905 160 1 a.2↑ (1000) 9 (27, 9) 0 (9, 9) 790 085 o 72 a.3↑ (2000) 9 (27, 9) 0 (9, 9) 494 236 total: 2 303

Table 11 characterises principalisation types of real quadratic fields with invariant 3 ≤ ν ≤ 4 by means of h3(L1) and Cl3(N1). Here we have always h3(Li) = 3 and Cl3(Ni) of type (3, 3) 2 for 2 ≤ i ≤ 4 and the second 3-class√ group G = Gal(F3(K)|K) is of coclass cc(G) = 1. Among 7 the 2 576 quadratic fields K = Q( D) with discriminant 0 < D < 10 and 3-class group Cl3(K) of type (3, 3) the dominating part of 2 303 fields, that is 89.4%, has at least a threefold total principalisation in the dihedral fields N1,...,N4. Table 12. Principalisation types with invariant 1 ≤ ν ≤ 2 for D > 0

1 type κ h3(L1) h3(L2) Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) ε Cl3(F3(K)) min. D fr. b.10 (0043) 9 9 (9, 9) (9, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 3, 3) 710 652 8 c.18 (0313) 9 3 (9, 9) (9, 3) (3, 3, 3) (9, 3) 1 (9, 3, 3) 534 824 29 c.21 (0231) 9 3 (9, 9) (9, 3) (9, 3) (9, 3) 0 (9, 3, 3) 540 365 25 d.19 (4043) 9 9 (27, 9) (9, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 3, 3) 2 328 721 1 d.23 (1043) 9 9 (27, 9) (9, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 3, 3) 1 535 117 1 c.21↑ (0231) 27 3 (27, 27) (9, 3) (9, 3) (9, 3) 0 (27, 9, 3) 1 001 957 2 d.25* (0143) 27 9 (27, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 9, 9, 3) 8 491 713 1 total: 67

Table 12 characterises principalisation types of real quadratic fields having invariant 1 ≤ ν ≤ 2 with the aid of h3(Li) for 1 ≤ i ≤ 2 and Cl3(Ni) for 1 ≤ i ≤ 4. For these cases we have h3(Li) = 3 2 for 3 ≤ i ≤ 4 and the second 3-class√ group G = Gal(F3(K)|K) is of coclass cc(G) ≥ 2. Among the 7 2 576 quadratic fields K = Q( D) with discriminant 0 < D < 10 and 3-class group Cl3(K) of type (3, 3) only a small part of 67 fields, that is 2.6%, has a single or double total principalisation in the dihedral fields N1,...,N4. Table 13 characterises the principalisation types of real quadratic fields having invariant ν = 0 by means of h3(Li) for 1 ≤ i ≤ 2 and Cl3(Ni) for 1 ≤ i ≤ 4. Here we have always h3(Li) = 3 for 2 3 ≤ i ≤ 4 and the second 3-class√ group G = Gal(F3(K)|K) is of coclass cc(G) ≥ 2. Among the 7 2 576 quadratic fields K = Q( D) with discriminant 0 < D < 10 and 3-class group Cl3(K) of type (3, 3), there is a modest part of 206 fields, that is 8.0%, which do not have a total principalisation in the dihedral fields N1,...,N4. Table 14 characterises the principalisation types of complex quadratic fields, which must have invariant ν = 0, by means of h3(Li) for 1 ≤ i ≤ 2 and Cl3(Ni) for 1 ≤ i ≤ 4. Here we have always 2 h3(Li) = 3 for 3 ≤ i ≤ 4 and the second√ 3-class group G = Gal(F3(K)|K) is of coclass cc(G) ≥ 2. There are 2 020 quadratic fields K = Q( D) with discriminant −106 < D < 0 and 3-class group Cl3(K) of type (3, 3). All these fields have exclusively partial principalisations in the dihedral fields N1,...,N4. PRINCIPALISATION ALGORITHM 21

Table 13. Second 3-class groups with invariant ν = 0 for D > 0

1 type κ h3(L1) h3(L2) Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) ε Cl3(F3(K)) min. D freq. D.5 (4224) 3 3 (3, 3, 3) (9, 3) (3, 3, 3) (9, 3) 2 (3, 3, 3) 631 769 47 D.10 (2241) 3 3 (9, 3) (9, 3) (3, 3, 3) (9, 3) 1 (3, 3, 3) 422 573 93 E.6 (1313) 9 3 (27, 9) (9, 3) (3, 3, 3) (9, 3) 1 (9, 9, 3) 5 264 069 o 7 E.14 (2313) 9 3 (27, 9) (9, 3) (3, 3, 3) (9, 3) 1 (9, 9, 3) 3 918 837 E.8 (1231) 9 3 (27, 9) (9, 3) (9, 3) (9, 3) 0 (9, 9, 3) 6 098 360 o 14 E.9 (2231) 9 3 (27, 9) (9, 3) (9, 3) (9, 3) 0 (9, 9, 3) 342 664 F.13 (3143) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 3) 8 321 505 1 G.16 (4231) 9 3 (27, 9) (9, 3) (9, 3) (9, 3) 0 (27, 9, 3) 8 711 453 2 G.19 (2143) 3 3 (9, 3) (9, 3) (9, 3) (9, 3) 0 (3, 3, 3, 3) 214 712 11 H.4 (4443) 3 3 (3, 3, 3) (3, 3, 3) (9, 3) (3, 3, 3) 3 (9, 3, 3) 957 013 27 F.13↑ (3143) 27 9 (81, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 27, 9, 3) 8 127 208 1 H.4↑ (3313) 9 3 (27, 9) (9, 3) (3, 3, 3) (9, 3) 1 (27, 9, 3) 1 162 949 3 total: 206

References [1] E. Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. Math. Sem. Univ. Hamburg 5 (1927), 353–363. [2] E. Artin, Idealklassen in Oberkörpern und allgemeines Reziprozitätsgesetz, Abh. Math. Sem. Univ. Hamburg 7 (1929), 46–51. [3] J. Ascione, G. Havas, and C. R. Leedham-Green, A computer aided classification of certain groups of prime power order, Bull. Austral. Math. Soc. 17 (1977), 257–274, Corrigendum 317–319, Microfiche Supplement p.320. [4] J. Ascione, On 3-groups of second maximal class (Ph.D. Thesis, Australian National University, Canberra, 1979). [5] J. Ascione, On 3-groups of second maximal class, Bull. Austral. Math. Soc. 21 (1980), 473–474. [6] G. Bagnera, La composizione dei gruppi finiti il cui grado èla quinta potenza di un numero primo, Ann. di Mat. (Ser. 3) 1 (1898), 137–228. [7] K. Belabas, Topics in computational algebraic number theory, J. Théor.Nombres Bordeaux 16 (2004), 19–63. [8] N. Blackburn, On a special class of p-groups, Acta Math. 100 (1958), 45–92. [9] J. R. Brink, The class field tower for imaginary quadratic number fields of type (3, 3) (Dissertation, Ohio State Univ., 1984). [10] H. Dietrich, B. Eick, and D. Feichtenschlager, Investigating p-groups by coclass with GAP, Computational group theory and the theory of groups, 45–61 (Contemp. Math. 470, AMS, Providence, RI, 2008). [11] B. Eick and C. Leedham-Green, On the classification of prime-power groups by coclass, Bull. London Math. Soc. 40 (2008), 274–288. [12] B. Eick and D. Feichtenschlager, Infinite sequences of p-groups with fixed coclass (preprint, 2010). [13] G. W. Fung and H. C. Williams, On the computation of a table of complex cubic fields with discriminant D > −106, Math. Comp. 55 (1990), nr. 191, 313–325. [14] Ph. Furtwängler,Beweis des Hauptidealsatzes fürdie Klassenkörper algebraischer Zahlkörper, Abh. Math. Sem. Univ. Hamburg 7 (1929), 14–36. [15] P. Hall, The classification of prime-power groups, J. Reine Angew. Math. 182 (1940), 130–141. [16] F.-P. Heider und B. Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Er- weiterungen, J. Reine Angew. Math. 336 (1982), 1–25. [17] R. James, The groups of order p6 (p an odd prime), Math. Comp. 34 (1980), nr. 150, 613–637. [18] H. Kisilevsky, Some results related to Hilbert’s theorem 94, J. Number Theory 2 (1970), 199–206. [19] P. Llorente and J. Quer, On totally real cubic fields with discriminant D < 107, Math. Comp. 50 (1988), nr. 182, 581–594. [20] D. C. Mayer, Principalization in complex S3-fields, Congressus Numerantium 80 (1991), 73–87 (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Winnipeg, Manitoba, Canada, 1990). [21] D. C. Mayer, List of discriminants dL < 200 000 of totally real cubic fields L, arranged according to their mul- tiplicities m and conductors f (Computer Centre, Department of Computer Science, University of Manitoba, Winnipeg, Canada, 1991). [22] D. C. Mayer, The second p-class group of a number field, in preparation. [23] D. C. Mayer, Transfers of metabelian p-groups, in preparation. [24] R. J. Miech, Metabelian p-groups of maximal class, Trans. Amer. Math. Soc. 152 (1970), 331–373. 22 DANIEL C. MAYER

Table 14. Second 3-class groups with invariant ν = 0 for D < 0

1 type κ h(L1) h(L2) Cl3(N1) Cl3(N2) Cl3(N3) Cl3(N4) ε Cl3(F3(K)) min.|D| freq. D.5 (4224) 3 3 (3, 3, 3) (9, 3) (3, 3, 3) (9, 3) 2 (3, 3, 3) 12 131 269 D.10 (2241) 3 3 (9, 3) (9, 3) (3, 3, 3) (9, 3) 1 (3, 3, 3) 4 027 667 E.6 (1313) 9 3 (27, 9) (9, 3) (3, 3, 3) (9, 3) 1 (9, 9, 3) 15 544 o 186 E.14 (2313) 9 3 (27, 9) (9, 3) (3, 3, 3) (9, 3) 1 (9, 9, 3) 16 627 E.8 (1231) 9 3 (27, 9) (9, 3) (9, 3) (9, 3) 0 (9, 9, 3) 34 867 o 197 E.9 (2231) 9 3 (27, 9) (9, 3) (9, 3) (9, 3) 0 (9, 9, 3) 9 748 F.7 (3443) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 3) 124 363 F.11 (1143) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 3) 27 156 o 78 F.12 (1343) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 3) 31 908 F.13 (3143) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 3) 67 480 G.16 (4231) 9 3 (27, 9) (9, 3) (9, 3) (9, 3) 0 (27, 9, 3) 17 131 79 G.19 (2143) 3 3 (9, 3) (9, 3) (9, 3) (9, 3) 0 (3, 3, 3, 3) 12 067 94 H.4 (4443) 3 3 (3, 3, 3) (3, 3, 3) (9, 3) (3, 3, 3) 3 (9, 3, 3) 3 896 297 E.6↑ (1313) 27 3 (81, 27) (9, 3) (3, 3, 3) (9, 3) 1 (27, 27, 3) 268 040 o 15 E.14↑ (2313) 27 3 (81, 27) (9, 3) (3, 3, 3) (9, 3) 1 (27, 27, 3) 262 744 E.8↑ (1231) 27 3 (81, 27) (9, 3) (9, 3) (9, 3) 0 (27, 27, 3) 370 740 o 13 E.9↑ (2231) 27 3 (81, 27) (9, 3) (9, 3) (9, 3) 0 (27, 27, 3) 297 079 F.7↑ (3443) 27 9 (81, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 27, 9, 3) 469 816 F.11↑ (1143) 27 9 (81, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 27, 9, 3) 469 787 o 14 F.12↑ (1343) 27 9 (81, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 27, 9, 3) 249 371 F.13↑ (3143) 27 9 (81, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 27, 9, 3) 159 208 G.16↑ (4231) 27 3 (81, 27) (9, 3) (9, 3) (9, 3) 0 (81, 27, 3) 819 743 2 H.4↑ (3313) 9 3 (27, 9) (9, 3) (3, 3, 3) (9, 3) 1 (27, 9, 3) 21 668 63 H.4↑2 (3313) 27 3 (81, 27) (9, 3) (3, 3, 3) (9, 3) 1 (81, 27, 3) 446 788 6 G.16r (1243) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 9, 9, 3) 290 703 ) G.19r (2143) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 9, 9, 3) 96 827 19 H.4r (3343) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (27, 9, 9, 3) 256 935 G.16i (1243) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 9) 135 059 ) G.19i (2143) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 9) 199 735 15 H.4i (3343) 9 9 (27, 9) (27, 9) (3, 3, 3) (3, 3, 3) 2 (9, 9, 9, 9) 186 483 F.12↑2 (1343) 27 27 (81, 27) (81, 27) (3, 3, 3) (3, 3, 3) 2 (27, 27, 27, 9) 423 640 1 G.19r↑ (2143) 27 9 (81, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (81, 27, 9, 3) 509 160 2 H.4r↑ (3343) 27 9 (81, 27) (27, 9) (3, 3, 3) (3, 3, 3) 2 (81, 27, 9, 3) 678 804 3 total: 2 020

[25] B. Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3, 3) und An- wendung auf das Kapitulationsproblem (Inauguraldissertation, Band 1, Universitätzu Köln,1989). [26] B. Nebelung, Anhang zu Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3, 3) und Anwendung auf das Kapitulationsproblem (Inauguraldissertation, Band 2, Universitätzu Köln,1989). [27] PARI/GP, version 2.3.4, http://pari.math.u-bordeaux.fr, Bordeaux, 2008. [28] A. Scholz, Idealklassen und Einheiten in kubischen Körpern, Monatsh. Math. Phys. 40 (1933), 211–222. [29] A. Scholz und O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginärquadratischer Zahlkörper: ihre rechnerische Bestimmung und ihr Einfluß auf den Klassenkörperturm, J. Reine Angew. Math. 171 (1934), 19–41. [30] O. Schreier, Uber¨ die Erweiterung von Gruppen II, Abh. Math. Sem. Univ. Hamburg 4 (1926), 321–346. [31] O. Taussky, A remark concerning Hilbert’s Theorem 94, J. Reine Angew. Math. 239/240 (1970), 435–438.

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