1.A. All known real quadratic examples:In the following diagrams, we give the 2stage metabelian 3groups G = G(K_{2}K) that occur for all real quadratic fields K = Q(d^{1/2}) with discriminant 0 < d < 200000 and 3class group of type (3,3). 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. 

Types a.3 and a.2
Remarks: The principalization types (0,1,0,0) and (1,0,0,0) do not determine the associated Galois group G = G(K_{2}K) uniquely. Results discovered 2003/02/07: However, according to top recent computations of the structure of Syl_{3}C(K_{1}) = G(K_{2}K_{1}) = G_{2} as (3,3) by Karim Belabas with the aid of PARI, G is determined uniquely for principalization type (1,0,0,0) and uniquely up to the sign of beta for principalization type (0,1,0,0). Further, exactly the same diagram illustrates the descending central series G = G_{1} >= G_{2} >= ... for the real quadratic fields K with discriminants d = 42817 of principalization type a.3, (0,1,0,0) [1,3] , resp. d = 94636 of principalization type a.2, (1,0,0,0) [1,3] . Results discovered 2003/02/27: In fact, we now have further occurrences of this frequent case: d = 103809, 114889, 130397, 142097, 151141, 153949, 172252, 173944, 184137, 189237, according to [3] and computations of Karim Belabas with the aid of PARI. For this frequent case, we obviously have generally Syl_{3}(C(K_{1})) = (3,3) <==> G(K_{2}K) in ZEF 2a(4,4) <==> family principal factorization type (Delta1,Alpha1,Alpha1,Alpha1) <==> principalization type either (1,0,0,0) or (0,1,0,0), i. e., the full 3class group of K capitulates in three of the four unramified cyclic cubic extensions N_{1},...,N_{4} of K, in the fourth one, say N_{1}, only a subgroup of type (3) capitulates and this is either Norm_{N_{1}K}(C(N_{1})) for type a.2, (1,0,0,0) or Norm_{N_{j}K}(C(N_{j})) with 2 <= j <= 4 for type a.3, (0,1,0,0). Further, the nonGalois cubic subfields L_{1},...,L_{4} of N_{1},...,N_{4} have uniformly 3class number 3. 

Type a.1
Remarks: Again, the principalization type (0,0,0,0) does not determine the associated Galois group G = G(K_{2}K) uniquely. Result discovered 2003/02/07: 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) by Karim Belabas with the aid of PARI, at least beta = 0, the class 5, and the order 729 of G are determined uniquely. Result discovered 2003/02/27: In fact, we now have two occurrences of this rare case: d = 62501 and d = 152949, according to [3] and a computation of Karim Belabas with the aid of PARI. For these fields, I got family principal factorization type (Alpha1,Alpha1,Alpha1,Alpha1) already in August 1991 at Winnipeg City [3] . For this rare case, we obviously have generally Syl_{3}(C(K_{1})) = (9,9) <==> G(K_{2}K) in ZEF 2b(6,6) <==> family principal factorization type (Alpha1,Alpha1,Alpha1,Alpha1) <==> principalization type a.1, (0,0,0,0), i. e., the full 3class group of K capitulates in all four unramified cyclic cubic extensions N_{1},...,N_{4} of K. Further, exactly one of the nonGalois cubic subfields L_{1},...,L_{4} of N_{1},...,N_{4} has 3class number 9, the others have 3class number only 3. This fact is due to formulas for the transfers ("Verlagerungen") and shows that the last parameter of the group G must be gamma = 1. 


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