1.A. All known real quadratic examples:In the following diagrams, we give the 2-stage metabelian 3-groups G = G(K2|K) that occur for all real quadratic fields K = Q(d1/2) with discriminant 0 < d < 200000 and 3-class group of type (3,3). We denote by G = G1 > G2 > ... > Gi > ... > Gm = 1 the descending central series of class m - 1 of G with Gi+1 = [Gi,G]. In particular, G2 = [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(K2|K) uniquely. Results discovered 2003/02/07: However, according to top recent computations of the structure of Syl3C(K1) = G(K2|K1) = G2 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 = G1 >= G2 >= ... 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 Syl3(C(K1)) = (3,3) <==> G(K2|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 3-class group of K capitulates in three of the four unramified cyclic cubic extensions N1,...,N4 of K, in the fourth one, say N1, only a subgroup of type (3) capitulates and this is either Norm_{N1|K}(C(N1)) for type a.2, (1,0,0,0) or Norm_{Nj|K}(C(Nj)) with 2 <= j <= 4 for type a.3, (0,1,0,0). Further, the non-Galois cubic subfields L1,...,L4 of N1,...,N4 have uniformly 3-class number 3. |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Type a.1
Remarks: Again, the principalization type (0,0,0,0) does not determine the associated Galois group G = G(K2|K) uniquely. Result discovered 2003/02/07: However, according to a top recent computation of the structure of Syl3C(K1) = G(K2|K1) = G2 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 Syl3(C(K1)) = (9,9) <==> G(K2|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 3-class group of K capitulates in all four unramified cyclic cubic extensions N1,...,N4 of K. Further, exactly one of the non-Galois cubic subfields L1,...,L4 of N1,...,N4 has 3-class number 9, the others have 3-class 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. |
||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
|
| <| Navigation Center <| |
| <| Back to Algebra <| |