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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), 1213-1237 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), 831-847 and S55-S58 |
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Web master's e-mail address:
contact@algebra.at |
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| Capitulation in Unramified Cubic Extensions of the Galois Closure of Pure Cubic Fields (2002/10/10) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
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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( R1/3 ) with class number hK coprime to 3 constitute 73885 of 827600, i. e., about 8.93% of all fields in the range R < 106 of normalized radicands R. The class number relation hN = hK2 * ui / 3 revealed that these fields can be characterized by the fact that the class number hN of the normal closure N of K is also coprime to 3. Thus, the overwhelming part of pure cubic fields ( > 91% ) has positive 3-class rank rK > 0. The investigations of Frank Gerth, George Gras, S. Kobayashi, and Thomas Callahan made it possible to analyze the 3-class groups of pure cubic fields with 3 | hK. |
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As a further step towards the investigation of positive 3-class rank rK > 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 3-groups 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 3-class group of N SN is of type (3,3) if and only if 3 || hK 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* = (K|Q)* of the cubic K over the rationals Q, and (**) the relative genus field N* = (N|k)* of the sextic N over the quadratic k, and characterizes them in terms of possible prime decompositions of the radicand R. |
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Since the 3-class group SN of N is elementary abelian of rank 2,
it contains exactly 4 subgroups H1,...,H4 of index (and order) 3, which correspond to intermediate fields F1,...,F4 of the Hilbert 3-class field N1 of N over N, because SN is isomorphic to the Galois group Gal( N1 | N ), by class field theory. Important subgroups of SN are: (#) the principal genus SND with D = 1 - sigma, where sigma denotes the generating automorphism of G = Gal( N | k ) = < sigma >, (##) the group of ambiguous classes (invariant under sigma), SNG, and (###) the eigenspaces SN+ and SN- to the eigenvalues 1 and -1 of the action of tau on SN, where tau denotes the generating automorphism of Gal( N | K ) = < tau >. Finally K1, K1', K1'' denote the Hilbert 3-class fields of the 3 conjugate absolutely cubic subfields K, K', K'' of N. By Ismaïli's condition 3 || hK, and by class field theory, Gal( K1 | K ) is isomorphic to SK of type C(3) (shortly (3)). In the following, we summarize the properties of Ismaïli's three configurations: |
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1st configuration:
SNG = SND = SN- are of type (3), and F4 = N* = NK1 = NK1' = NK1'' coincide, all together. ker( j_(F4|N) ) = SN, i. e., all 3-classes of N become principal in F4 = 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* = K1, by a result of Ishida. Possible radicands are: (*) R = p = 1 (mod 3), (**) R = 3p,9p with p = 4,7 (mod 9), (***) R = pq,p2q,pq2 = 1,8 (mod 9) with p = 4,7 (mod 9) or q = 2,5 (mod 9).
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2nd configuration:
SNG = SND = SN+ = SK are of type (3), and F1 = N*, F2 = NK1'', F3 = NK1', and F4 = NK1 are all distinct. For all i = 1,...,4, ker( j_(Fi|N) ) contains SNG = H1, i. e., the ambiguous 3-classes of N become principal in all 4 intermediate fields of N1 | 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 = q1q2,q12q2 with q1,q2 = 8 (mod 9), (***) R = q1xq2y,3q1xq2y,9q1xq2y incongruent 1,8 (mod 9) with x,y = 1 or 2, q1 = 2,5 (mod 9) or q2 = 2,5 (mod 9), (****) R = q1xq2yq3z = 1,8 (mod 9) with x,y,z = 1 or 2 and q1 = 2,5 (mod 9) or q2 = 2,5 (mod 9) or q3 = 2,5 (mod 9).
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3rd configuration:
This is the most complex case. For this case, the results of Ismaïli's thesis have been refined in [2] . The principal genus SND = 1 is trivial, i. e., coincides with the principal class, whence all 3-classes of N are ambiguous, SNG = SN is of type (3,3), and N* = N1. Further, F4 = NK1 = NK1' = NK1'' coincide. ker( j_(F4|N) ) = SN, i. e., all 3-classes of N become principal in F4 = NK1, 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* = K1, by a result of Ishida, and NK1 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,p2q,pq2 with p = 1 (mod 9), q = 8 (mod 9), (***) R = pxqy,3pxqy,9pxqy incongruent 1,8 (mod 9) with x,y = 1 or 2, p = 4,7 (mod 9) or q = 2,5 (mod 9), (****) R = pxq1yq2z = 1,8 (mod 9) with x,y,z = 1 or 2 and p = 4,7 (mod 9) or q1 = 2,5 (mod 9) or q2 = 2,5 (mod 9).
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More information about the possible conductors f and radicands R
occuring in Ismaïli's 3 configurations is summarized in some tables of details . |
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References:
[0] Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989 [1] Moulay Chrif Ismaïli, Sur la capitulation des 3-classes 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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