Principalization Types of

2-Stage Metabelian 3-Groups

1. The Connection between Class, Order, and Commutator Subgroup. In our previous communication, concerning the descending central series of 2-stage metabelian 3-groups, we published several top recent results (discovered between January and June, 2003) for quadratic fields K whose Sylow 3-subgroup Syl3(C(K)) of the class group C(K), i. e., the 3-class group, is of type (3,3). We denoted by G = G1 >= G2 >= ... >= Gi >= ... the descending central series of G with Gi+1 = [Gi,G]. In particular, G2 = [G,G] is the commutator subgroup G' of G. Now it is adequate to give the solution of a very fundamental problem: How do we get the class m-1 and the order 3n of the 2-stage metabelian 3-group G = G(K2|K) in ZEF(m,n) of the 2nd Hilbert 3-class field K2 of an arbitrary base field K with 3-class group Syl3(C(K)) = (3,3) over K from the structure of the (abelian) commutator subgroup G' = G2 = Gal(K2|K1) = Syl3(C(K1)) which coincides with the 3-class group of the 1st Hilbert 3-class field K1 of K: 1. There are bounds for the order in dependence on the class: m <= n <= 2m - 3 2. Since there are only upper triangular entries in an (m,n)-diagram, we use an ordering parameter e = n + 2 - m ( >= 2 ) to measure the distance from the diagonal ( e = 2 ) 3. Then we have an almost unique correspondence between the structure of G' and the pair (m,n), expressed in the following (easily extensible) table . Only the red pairs (m,n) are exceptional, since they correspond to 2 possible structures of G'. In particular, we get 3 theorems concerning important cases: THEOREM 1. G is of maximal class ( n = m, resp. e = 2 ) <==> the 3-rank of G' is r <= 2 THEOREM 2. G is of second maximal class ( n = m + 1, resp. e = 3 ) <==> the 3-rank of G' is r = 3 or G' = (3,3,3,3) THEOREM 3. G is of lower than second maximal class ( n >= m + 2, resp. e >= 4 ) <==> the 3-rank of G' is r = 4 and the order of G' is >= 243

2. The Connection between Principalization Types, Class, and Order. The next question is: How are the principalization types related to the parameter e and the pairs (m,n) ? The fine (necessary and sufficient) conditions have been determined by Brigitte Nebelung [4] . Here, we only give a coarse connection: Denote by 0 <= f <= 4 the number of those unramified cubic extensions of K where the full 3-class group Syl_3(C(K)) = (3,3) of K becomes principal (i. e., capitulates). Then we have 2 theorems: THEOREM (i) G is of maximal class ( n = m, resp. e = 2 ) <==> f = 4, i. e., principalization type (0,0,0,0) or f = 3, i. e., principalization type (1,0,0,0) or (2,0,0,0) or f = 0 and the principalization type is (1,1,1,1) (which implies n = 3 ) THEOREM (ii) G is of second maximal class or lower ( n >= m + 1, resp. e >= 3 ) <==> f <= 2 but the principalization type is different from (1,1,1,1)

3. Peculiarities of Quadratic Fields. The last question is, what happens, if K is a quadratic field ? Well, according to Scholz and Taussky [1] we have 2 theorems: THEOREM 1. Principalization type (1,1,1,1) (denoted as Type "A" by S. & T.) is impossible for quadratic fields K THEOREM 2. f = 0 for any imaginary quadratic field K Example: d = -28031 ==> f = 0 and p.t. != (1,1,1,1) ==> e != 2 ==> r >= 3 ==> G' != (9,3). In fact, G' = (3,3,3).

4. Summary of recently (2003) discovered connections. In general, Nebelung's results still admit a wide range of 2-stage metabelian 3-groups G = G(K2 | K) for the 2nd Hilbert 3-class field K2 of an algebraic number field K with 3-class group Syl3(C(K)) of type (3,3) and with a certain principalization type (i1,...,i4) in the four unramified cyclic cubic extensions N1,...,N4 of K. However, our concrete numerical results [7] discovered in 2003 for quadratic base fields K show that, in this special case, a principalization type (i1,...,i4) uniquely determines the class m-1 and order 3n of G = G(K2 | K), except in the sections "G" and "H", where two possibilities arise. (Section "A" is impossible for quadratic base fields K and sections "B" and "C" cannot occur at all, for group theoretic reasons.) In the following table, the types (i1,...,i4) are arranged into sections, according to [1,4] , and they are numbered similarly as in [5] . We always give a canonical representative (CR) of the type's equivalence class, the year of the concrete numerical realization of the type with a reference, the first discriminant dK of a quadratic field K with that type, the number of fixed points (FP), the occupation numbers (ON) (telling how often each of the digits 1,2,3,4 appears in the representative), the cardinality (#) of the type's orbit under the operation of S4, an ideal of polynomials in Z[X,Y], called the associated symbolic order (SO) in [1] , the structure of the commutator subgroup G' = G2, the exponents in defining relations for the group's generators (RE), and the set ZEF(m,n), defined in [4] , to which the group G belongs. Finally, we note that the family of 3-class numbers (h1,...,h4) of the absolute cubic subfields L1,...,L4 of the normal S3-fields N1,...,N4 between K1 and K also permits partial conclusions concerning the group G. Sec. No. CR Year dK FP ON # SO h1,...,h4 G2 RE G in a 1 2000 1981 [2] 32009 3 - 12 ? 3,3,3,3 (3,3) 0,+-1,0 ZEF 2a(4,4) 2 1000 1981 [2] 72329 4 - 4 ? 3,3,3,3 (3,3) 1,0,0 ZEF 2a(4,4) 3 0000 1981 [2] 62501 4 - 1 ? 9,3,3,3 (9,9) ?,0,1 ZEF 2b(6,6) [A 1 1111 1931 [0] - 1 4000 4 L - (3) 1,0,0 ZEF 1a(3,3)] B 2 1211 2 3100 12 3 1112 1 3100 24 C 15 1234 4 1111 1 17 1342 1 1111 8 18 2341 0 1111 6 D 5 1212 1981 [2] -12131 2 2200 12 L2 3,3,3,3 (3,3,3) 1,1,-1,1;0 ZEF 1a(4,5) 10 1123 1933 [1] -4027 1 2110 24 L2 3,3,3,3 (3,3,3) 0,0,-1,1;0 ZEF 1a(4,5) E 6 1122 1981 [2] -15544 1 2200 12 X4 9,3,3,3 (9,9,3) 1,-1,1,1;0 ZEF 2a(6,7) 8 1231 2003 [7] -34867 3 2110 12 X4 9,3,3,3 (9,9,3) 1,0,-1,1;0 ZEF 2a(6,7) 9 1132 1933 [1] -9748 2 2110 24 X4 9,3,3,3 (9,9,3) 0,0,+-1,1;0 ZEF 2a(6,7) 14 2311 1981 [2] -16627 0 2110 24 X4 9,3,3,3 (9,9,3) 0,-1,+-1,1;0 ZEF 2a(6,7) F 7 2112 2003 [7] -124363 0 2200 12 Ra,b 9,9,3,3 (9,9,9,3) ?,b,?,b;0 ZEF 1a(6,9) 11 1321 1989 [5] -27156 1 2110 12 Ra,b 9,9,3,3 (9,9,9,3) ?,?,0,0;0 ZEF 1a(6,9) 12 2131 2003 [7] -31908 1 2110 24 Ra,b 9,9,3,3 (9,9,9,3) ?,?,0,?;0 ZEF 1a(6,9) 13 2113 0 2110 24 2003 [7] -67480 Ra,b 9,9,3,3 (9,9,9,3) ?,?,?,0;0 ZEF 1a(6,9) 2003 [7] -159208 Ra,b 27,9,3,3 (27,27,9,3) ?,?,?,0;0 ZEF 2a(8,11) G 16 1243 2 1111 6 1981 [2] -17131 X5? 9,3,3,3 (27,9,3) ?,0,?,1;+-1 ZEF 2b(7,8) 2003 [7] -135059 ? 9,9,3,3 (9,9,9,9) ?,?,?,0;-1 ZEF 1b(7,10) 19 2143 0 1111 3 1981 [2] -12067 ? 3,3,3,3 (3,3,3,3) ?,0,?,1;-1 ZEF 1b(5,6) 2003 [7] -96827 ? 9,9,3,3 (27,9,9,3) ?,?,?,0;+-1 ZEF 1b(7,10) H 4 2111 0 3100 12 1981 [2] -3896 X3? 3,3,3,3 (9,3,3) ?,-1,?,1;+-1 ZEF 1b(5,6) 2003 [7] -21668 X5? 9,3,3,3 (27,9,3) ?,-1,?,1;+-1 ZEF 2b(7,8)

Gratifications. The year 2003 brought the hard boiled types that were missing for more than 20 years since [2]. I definitely felt much internal inspiration and motivation to complete this task by the spirits of the genii Arnold Scholz (1904 - 1942) and Alexander Aigner (1909 - 1988) and I express my respect and gratitude to them for helping me to succeed in my final enterprise. However, in spite of this metaphysical energy, I wouldn't have been able to gain such deep insight without the concrete numerical aid concerning number fields of absolute degree 18 and their class group structure by Karim Belabas at Orsay, Paris, France.

References: [0] Olga Taussky, Über eine Verschärfung des Hauptidealsatzes für algebraische Zahlkörper, J. reine angew. Math.168 (1932), 193 - 210 [1] Arnold Scholz und Olga Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. reine angew. Math.171 (1934), 19 - 41 [2] Franz-Peter Heider und Bodo Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25 [3] James R. Brink and Robert Gold, Class field towers of imaginary quadratic fields, manuscripta math. 57 (1987), 425 - 450 [4] Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989 [5] Daniel C. Mayer, Principalization in complex S3-fields, Congressus Numerantium 80 (1991), 73 - 87 [6] Daniel C. Mayer, List of discriminants dL<200000 of totally real cubic fields L, arranged according to their multiplicities m and conductors f, 1991, Dept. of Comp. Sci., Univ. of Manitoba [7] Daniel C. Mayer, Principalization in Unramified Cubic Extensions of Quadratic Fields with Discriminant -50000 < d < 0, Univ. Graz, Computer Centre, 2003