1.B. All known imaginary quadratic examples:In the following diagrams, we give the 2stage metabelian 3groups G = G(K_{2}K) that occur for all imaginary quadratic fields K = Q(d^{1/2}) with discriminant 50000 < d < 0 and 3class group of type (3,3). These recent computations of 2003 for 42 new cases in the range 50000 < d < 30000 [6] extend our own results for 22 fields with 30000 < d < 20000 of 1989 [5] and the 13 examples with 20000 < d < 0 of Heider and Schmithals in 1982 [2] . As a supplement, we give some singular cases with groups of exceptionally high order in the range 200000 < d < 50000. We denote by G = G_{1} >= G_{2} > ... > G_{i} > ... > G_{m} = 1 the descending central series of class m  1 of G with G_{i+1} = [G_{i},G]. In particular, G_{2} = [G,G] is the commutator subgroup G' of G. 

Type D36 of 77 fields (47 %)14 (39 %) of Type D.5 and 22 (61 %) of Type D.10
Remarks: (1,1,2,3) and (1,2,1,2) are the only two principalization types where the associated Galois group G = G(K_{2}K) is uniquely determined. Exactly the same diagram illustrates the descending central series G = G_{1} >= G_{2} >= ... for the imaginary quadratic fields K with discriminants d = 8751, 19651, 21224, 22711, 24904, 26139, 28031, 28759 of principalization type D.10, (1,1,2,3) [2,5] , resp. d = 19187, 20276, 20568, 24340, 26760 of principalization type D.5, (1,2,1,2) [2,5] . The Hilbert 3class field tower K = K_{0} <= K_{1} <= K_{2} <= ... of all these fields terminates after 2 steps with K_{2}, i. e., 3 does not divide the class number of K_{2}. This was shown by Scholz and Taussky in [1] and, with a different proof, by Brink and Gold in [3] . Results discovered 2003/04/19  22 and 2003/05/09: According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case: d = 31639, 31999, 32968, 34507, 35367, 41583, 41671, 43307 of principalization type D.5, (1,2,1,2) and d = 34088, 36807, 40299, 40692, 41015, 42423, 43192, 44004, 45835, 46587, 48052, 49128, 49812 of principalization type D.10, (1,1,2,3). 

Type E18 of 77 fields (23 %)3 (17 %) of Type E.6, 2 (11 %) of Type E.8, 9 (50 %) of Type E.9, and 4 (22 %) of Type E.14
Remarks: Here, the principalization type (1,1,3,2) does not determine the associated Galois group G = G(K_{2}K) uniquely. However, Scholz and Taussky [1] provide additional information for d = 9748 mentioning that a = 4 in the associated symbolic order X_{a} = (X^{a},XY,Y^{2},3+3X+3X^{2}) whence G' = G_{2} = Z[X,Y]/X_{a} = (3^{2},3^{2},3), and this property determines G uniquely, up to the sign of gamma. The Hilbert 3class field tower K = K_{0} <= K_{1} <= K_{2} <= ... of the field K = Q((9748)^{1/2}) terminates after 2 steps with K_{2}, i. e., 3 does not divide the class number of K_{2}. That is the only claim by Scholz and Taussky in [1] and not a statement for all fields with associated symbolic order X_{a}. Heider and Schmithals erroneously remark in [2] that Scholz and Taussky proved G_{4} = 1 for the fields K with associated symbolic order X_{a}, whereas they really have G_{4} = (3,3). This led to a misinterpretation by Brink and Gold in [3] , where they construct a 3stage metabelian 3group that could possibly be the Galois group G(MK) of an unramified cyclic cubic extension M of K_{2} when K has the associated symbolic order X_{a}. But no explicit example is known up to now for a 3class field tower of height greater than 2. Results discovered 2003/01/14  18: According to recent computations of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (9,9,3) by Karim Belabas with the aid of PARI, exactly the same diagram illustrates the descending central series G = G_{1} >= G_{2} >= ... for the imaginary quadratic fields K with discriminants d = 22395, 22443, 27640 of principalization type E.9, (1,1,3,2) ~ (1,2,1,3) [5] , with (alpha,beta,gamma,delta) = (0,0,+1,1), rho = 0, resp. d = 15544, 18555, 23683 of principalization type E.6, (1,1,2,2) [2,5] , with (alpha,beta,gamma,delta) = (1,1,1,1), rho = 0, resp. d = 16627 of principalization type E.14, (2,3,1,1) [2,5] , with (alpha,beta,gamma,delta) = (0,1,+1,1), rho = 0, which are all designated by E in [1] . Results discovered 2003/03/23 and 2003/05/24: According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case: d = 31271, 34867, 37988, 39736, 42619, 42859, 43847, 45887, 48472, 48667. More detailed, according to [6] , we finally have d = 37988, 39736, 45887, 48472, 48667 of principalization type E.9, (1,1,3,2) ~ (1,2,1,3), with (alpha,beta,gamma,delta) = (0,0,+1,1), rho = 0, d = 31271, 42859, 43847 of principalization type E.14, (2,3,1,1), with (alpha,beta,gamma,delta) = (0,1,+1,1), rho = 0, and d = 34867, 42619 of the newly discovered principalization type E.8, (1,2,3,1), the unique one with 3 fixed points (A,A,A,B) and with (alpha,beta,gamma,delta) = (1,0,1,1), rho = 0. 

Type H13 of 77 fields (17 %)
Remarks: Again, the principalization type (2,1,1,1) does not determine the associated Galois group G = G(K_{2}K) uniquely. Results discovered 2003/01/14 and 2003/02/17: However, according to recent computations of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (9,3,3) by Karim Belabas with the aid of PARI and Claus Fieker with the aid of MAGMA, at least delta = 1, the class 4, and the order 729 of G are determined uniquely. Exactly the same diagram illustrates the descending central series G = G_{1} >= G_{2} >= ... for the imaginary quadratic fields K with discriminants d = 6583, 23428, 25447, 27355, 27991 of the same principalization type H.4, (2,1,1,1) [2,5] , designated by H in [1] . Results discovered 2003/04/22 and 2003/05/09: According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case: d = 36276, 37219, 37540, 39819, 41063, and: d = 43827, 46551. 

Type G.192 of 77 fields (3 %)
Remarks: As above, the principalization type (2,1,4,3) does not determine the associated Galois group G = G(K_{2}K) uniquely. Result discovered 2003/01/14: But according to the recent computation of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (3,3,3,3) by Karim Belabas with the aid of PARI, at least beta = 0, delta = 1, rho = 1, the class 4, and the order 729 of G are determined uniquely. Result discovered 2003/05/10: According to [6] , we now have another occurrence of this rare case: d = 49924. 

Type F2 of 77 fields (3 %)1 (50 %) of Type F.11, 1 (50 %) of Type F.12
Remarks: Again, the principalization type (1,3,2,1) does not determine the associated Galois group G = G(K_{2}K) uniquely. Result discovered 2003/02/13: However, according to a top recent computation of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (9,9,9,3) by Claus Fieker with the aid of MAGMA, at least gamma = 0, delta = 0, rho = 0, the class 5, and the order 19683 of G are determined uniquely. Results discovered 2003/03/23, 2003/05/27  31, and 2003/06/10: According to [6] and computations of Karim Belabas with the aid of PARI, we now have other occurrences of this rare case: d = 31908, 135587 of the newly discovered principalization type F.12, (2,1,3,1) ~ (3,2,1,1). According to the supplements section [7] of [6] , we finally have other occurrences of this rare case: d = 67480, 104627 of the newly discovered principalization type F.13, (2,1,1,3), and d = 124363 of the newly discovered principalization type F.7, (2,1,1,2). Results discovered 2003/09/16: According to the supplements section [7] of [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case: d = 160403, 184132, 189959 of type F.12, (2,1,3,1) ~ (3,2,1,1) and d = 167064 of type F.13, (2,1,1,3). 

Type G.16 and a Variant of Type H4 of 77 fields (5 %) ... Type G.162 of 77 fields (3 %) ... Variant of Type H.4
Remarks: As above, the principalization type (1,2,4,3) does not determine the associated Galois group G = G(K_{2}K) uniquely. Results discovered 2003/02/17: However, according to a top recent computation of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (27,9,3) by Claus Fieker with the aid of MAGMA, at least beta = 0, delta = 1, the class 6, and the order 6561 of G are determined uniquely. Exactly the same diagram illustrates the descending central series G = G_{1} >= G_{2} >= ... for the imaginary quadratic fields K with discriminants d = 24884, 28279 of the same principalization type G.16, (1,2,4,3) ~ (2,1,3,4) [5] , with (alpha,beta,gamma,delta) = (1,0,0,1), rho = 0 or (alpha,beta,gamma,delta) = (?,0,?,1), rho = +1, designated by G in [1] , resp. d = 21668 of principalization type H.4.V1, (2,1,1,1) [5] , with (alpha,beta,gamma,delta) = (1,1,1,1), rho = 0 or (alpha,beta,gamma,delta) = (?,1,?,1), rho = +1, designated by H in [1] . Results discovered 2003/03/23: According to [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case: d = 34027 of principalization type H.4.V1, (2,1,1,1) and d = 35539 of principalization type G.16, (1,2,4,3) ~ (2,1,3,4). 

A Variant of Type G.19
Remarks: As above, the principalization type (2,1,4,3) does not determine the associated Galois group G = G(K_{2}K) uniquely. Result discovered 2003/05/29: However, according to a top recent computation of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (27,9,9,3) by Karim Belabas with the aid of PARI, at least delta = 0, the class 6, and the order 59049 of G are determined uniquely. Result discovered 2003/09/16: According to the supplements section [7] of [6] and a computation of Karim Belabas with the aid of PARI, we now have a further occurrence of this case: d = 156452. 

A Variant of Type G.16 and another Variant of Type H
Remarks: As above, the principalization type (1,2,4,3) does not determine the associated Galois group G = G(K_{2}K) uniquely. Result discovered 2003/06/10: However, according to a top recent computation of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (9,9,9,9) by Karim Belabas with the aid of PARI, at least delta = 0, rho = 1, the class 6, and the order 59049 of G are determined uniquely. Results discovered 2003/09/16 and 2004/01/06: According to the supplements section [7] of [6] and computations of Karim Belabas with the aid of PARI, we now have further occurrences of this case: d = 185747 of type G.16.V, (1,2,4,3), and d = 186483 of type H.4.V2, (2,1,1,1). 

A Variant of Type F.13
Remarks: As above, the principalization type (2,1,1,3) does not determine the associated Galois group G = G(K_{2}K) uniquely. Result discovered 2003/08/10: However, according to an investigation of the transfers ("Verlagerungen") to the 4 maximal subgroups M_{1},...,M_{4} of G, at least delta = 0, rho = 0, the class 7, and the order 177147 of G are determined uniquely. 


< Navigation Center < 
< Back to Algebra < 