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On these pages, we present most recent results of our joint research, directly from the lab. 
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Basic bibliography:
K. Belabas, A fast algorithm to compute cubic fields, Math. Comp. 66 (1997), 12131237 A. Derhem, Capitulation dans les extensions quadratiques non ramifiées de corps de nombres cubiques cycliques, Thèse de doctorat, Université Laval, Québec, 1988 D. C. Mayer, Multiplicities of dihedral discriminants, Math. Comp. 58 (1992), 831847 and S55S58 
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Web master's email address:
contact@algebra.at 
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Capitulation in Unramified Cubic Extensions of the Galois Closure of Pure Cubic Fields (2002/10/10)  
Dan (02/10/10):
In my last communication , whose notation I use throughout the present article, we have seen that pure cubic fields K = Q( R^{1/3} ) with class number h_{K} coprime to 3 constitute 73885 of 827600, i. e., about 8.93% of all fields in the range R < 10^{6} of normalized radicands R. The class number relation h_{N} = h_{K}^{2} * ui / 3 revealed that these fields can be characterized by the fact that the class number h_{N} of the normal closure N of K is also coprime to 3. Thus, the overwhelming part of pure cubic fields ( > 91% ) has positive 3class rank r_{K} > 0. The investigations of Frank Gerth, George Gras, S. Kobayashi, and Thomas Callahan made it possible to analyze the 3class groups of pure cubic fields with 3  h_{K}. 

As a further step towards the investigation of positive 3class rank r_{K} > 0,
Moulay Chrif Ismaïli studies in his 1992 thesis [1] the capitulation in unramified cubic extensions of the Galois closure N = K( zeta ) of pure cubic fields. Up to now, the theoretical framework for investigations of this kind is the classification of metabelian 3groups with commutator factor group of type C(3)*C(3) (shortly (3,3)) in the 1989 thesis [0] of Brigitte Nebelung, which extends the foundations of Scholz and Taussky. Hence, Ismaïli starts by proving that the 3class group of N S_{N} is of type (3,3) if and only if 3  h_{K} and ui = 3. In particular, unit index ui = 3 implies that all the fields he considers are of Principal Factorization Type BETA or GAMMA. Then Ismaïli distinguishes 3 typical configurations of genus fields, (*) the absolute genus field K* = (KQ)* of the cubic K over the rationals Q, and (**) the relative genus field N* = (Nk)* of the sextic N over the quadratic k, and characterizes them in terms of possible prime decompositions of the radicand R. 

Since the 3class group S_{N} of N is elementary abelian of rank 2,
it contains exactly 4 subgroups H_{1},...,H_{4} of index (and order) 3, which correspond to intermediate fields F_{1},...,F_{4} of the Hilbert 3class field N_{1} of N over N, because S_{N} is isomorphic to the Galois group Gal( N_{1}  N ), by class field theory. Important subgroups of S_{N} are: (#) the principal genus S_{N}^{D} with D = 1  sigma, where sigma denotes the generating automorphism of G = Gal( N  k ) = < sigma >, (##) the group of ambiguous classes (invariant under sigma), S_{N}^{G}, and (###) the eigenspaces S_{N}^{+} and S_{N}^{} to the eigenvalues 1 and 1 of the action of tau on S_{N}, where tau denotes the generating automorphism of Gal( N  K ) = < tau >. Finally K_{1}, K_{1}', K_{1}'' denote the Hilbert 3class fields of the 3 conjugate absolutely cubic subfields K, K', K'' of N. By Ismaïli's condition 3  h_{K}, and by class field theory, Gal( K_{1}  K ) is isomorphic to S_{K} of type C(3) (shortly (3)). In the following, we summarize the properties of Ismaïli's three configurations: 

1^{st} configuration:
S_{N}^{G} = S_{N}^{D} = S_{N}^{} are of type (3), and F_{4} = N* = NK_{1} = NK_{1}' = NK_{1}'' coincide, all together. ker( j_(F_{4}N) ) = S_{N}, i. e., all 3classes of N become principal in F_{4} = N*, and we have 3 possible capitulation types: (0,0,0,0), (1,2,3,0), (4,4,4,0). There exists a prime p = 1 (mod 3) with p  R, whence K* = K_{1}, by a result of Ishida. Possible radicands are: (*) R = p = 1 (mod 3), (**) R = 3p,9p with p = 4,7 (mod 9), (***) R = pq,p^{2}q,pq^{2} = 1,8 (mod 9) with p = 4,7 (mod 9) or q = 2,5 (mod 9).


2^{nd} configuration:
S_{N}^{G} = S_{N}^{D} = S_{N}^{+} = S_{K} are of type (3), and F_{1} = N*, F_{2} = NK_{1}'', F_{3} = NK_{1}', and F_{4} = NK_{1} are all distinct. For all i = 1,...,4, ker( j_(F_{i}N) ) contains S_{N}^{G} = H_{1}, i. e., the ambiguous 3classes of N become principal in all 4 intermediate fields of N_{1}  N, and we have 4 possible capitulation types: (0,0,0,0), (0,1,1,1), (1,0,0,0), (1,1,1,1). For all primes q != 3 with q  R, we have q = 2 (mod 3), whence K* = K, by a result of Ishida. Possible radicands are: (*) R = 3q,9q with q = 8 (mod 9), (**) R = q_{1}q_{2},q_{1}^{2}q_{2} with q_{1},q_{2} = 8 (mod 9), (***) R = q_{1}^{x}q_{2}^{y},3q_{1}^{x}q_{2}^{y},9q_{1}^{x}q_{2}^{y} incongruent 1,8 (mod 9) with x,y = 1 or 2, q_{1} = 2,5 (mod 9) or q_{2} = 2,5 (mod 9), (****) R = q_{1}^{x}q_{2}^{y}q_{3}^{z} = 1,8 (mod 9) with x,y,z = 1 or 2 and q_{1} = 2,5 (mod 9) or q_{2} = 2,5 (mod 9) or q_{3} = 2,5 (mod 9).


3^{rd} configuration:
This is the most complex case. For this case, the results of Ismaïli's thesis have been refined in [2] . The principal genus S_{N}^{D} = 1 is trivial, i. e., coincides with the principal class, whence all 3classes of N are ambiguous, S_{N}^{G} = S_{N} is of type (3,3), and N* = N_{1}. Further, F_{4} = NK_{1} = NK_{1}' = NK_{1}'' coincide. ker( j_(F_{4}N) ) = S_{N}, i. e., all 3classes of N become principal in F_{4} = NK_{1}, and there are 4 possible capitulation types: (0,0,0,0), (0,3,2,0), (1,0,0,0), (1,3,2,0), provided the radicand is of the shape (*) or (**) or (***) with p = 1 (mod 9) or (***) with p = 4,7 (mod 9) and R coprime to 3 or (****) with p = 1 (mod 9). The capitulation for the other radicands is more complicated. There exists a prime p = 1 (mod 3) with p  R, whence K* = K_{1}, by a result of Ishida, and NK_{1} is the compositum of N with L(p) where L(p) denotes the cyclic cubic field with conductor p. Possible radicands are: (*) R = 3p,9p with p = 1 (mod 9), (**) R = pq,p^{2}q,pq^{2} with p = 1 (mod 9), q = 8 (mod 9), (***) R = p^{x}q^{y},3p^{x}q^{y},9p^{x}q^{y} incongruent 1,8 (mod 9) with x,y = 1 or 2, p = 4,7 (mod 9) or q = 2,5 (mod 9), (****) R = p^{x}q_{1}^{y}q_{2}^{z} = 1,8 (mod 9) with x,y,z = 1 or 2 and p = 4,7 (mod 9) or q_{1} = 2,5 (mod 9) or q_{2} = 2,5 (mod 9).


More information about the possible conductors f and radicands R
occuring in Ismaïli's 3 configurations is summarized in some tables of details . 
References:
[0] Brigitte Nebelung, Klassifikation metabelscher 3Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989 [1] Moulay Chrif Ismaïli, Sur la capitulation des 3classes d'idéaux de la clôture normale d'un corps cubique pur, Thèse de doctorat, Université Laval, Québec, 1992 [2] Moulay Chrif Ismaïli and Rachid El Mesaoudi, Sur la divisibilité exacte par 3 du nombre de classes de certain corps cubiques purs, Ann. Sci. Math. Québec 25 (2001), no. 2, 153  177 
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