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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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Email addresses:
Karim.Belabas@math.upsud.fr aderhem@yahoo.fr danielmayer@algebra.at 
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Capitulation in Unramified Cubic Extensions of Cyclic Cubic Fields (2002/07/18) 
Dan (02/07/18):
For a cyclic cubic field k with 2prime conductor f = p*q, where p = 1(mod 3) or p = 3^{2} and q = 1(mod 3) and where we put p' = 3 if p = 3^{2} and p' = p otherwise, we study the Hilbert 3class field tower, k = k_{0} <= k_{1} <= k_{2} <= ... <= k_{n} <= ..., the absolute genus field k_{0} <= k* <= k_{1}, and the possible capitulation types, when the 3class group of k is of type C_{3} or C_{3}*C_{3}. G denotes the absolute Galois group Gal( k*_{1}  Q ) of the Hilbert 3class field k*_{1} of the genus field k* [4] and G = G_{1} >= G_{2} >= ... >= G_{i} >= ... the descending central series [1] of G with G_{i+1} = [G_{i},G]. In particular, G_{2} = [G,G] is the commutator subgroup of G. The genus field k* is the compositum k* = k_{p}*k_{q} of k_{p} and k_{q}, the cyclic cubics with prime conductors p and q. Hence k* is a bicyclic bicubic field [2] and contains another cyclic cubic k' with conductor f = p*q. The unit group U of k* contains the subgroup V of old units generated by all units of the subfields k, k', k_{p}, and k_{q}. We put H = Gal( k*_{1}  k ) and H' = Gal( k*_{1}  k' ). 
1. The fields k with 3class rank rho = 1
and 3class group C_{3}, for which p and q are not mutual cubic residues. Here the genus field k* has 3class number h = 1 and unit group index (U:V) = 27. If (p/q)_{3} < > 1 and (q/p)_{3} < > 1, then the principal factors of k and of k' are either { p'q, p'^{2}q^{2} } or { p'^{2}q, p'q^{2} } but they differ for k and k'. If (p/q)_{3} = 1 and (q/p)_{3} < > 1, then the principal factors of k and k' coincide and are { p', p'^{2} } [3,5] . G = C_{3}*C_{3} in ZEF(2,2), the abelian 3group (of maximal class 1) [1,4] of order 9, and exponent 3. Since H and H' = C_{3}, the capitulation type of k and of k' is trivial, i. e., the (full) 3class groups C_{3} of k and k' become principal in k*, as Hilbert's Theorem 94 asserts in this simple case. The Hilbert 3class field tower terminates after one step [3] . 



The smallest examples of C(k) = C(k') = C_{3} are
[5]
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a) f = 63 = 3^{2}*7 with (9/7)_{3} < > 1 and (7/9)_{3} < > 1 and with distinct principal factors { 3^{2}*7, 3*7^{2} } in k and { 3*7, 3^{2}*7^{2} } in k'. b) f = 91 = 7*13 with (13/7)_{3} = 1 but (7/13)_{3} < > 1 and with coinciding principal factors { 13, 13^{2} } in k and k'. 
2. The fields k with 3class rank rho = 2
and 3class group C_{3}*C_{3}, for which p and q are cubic residues with respect to each other. 

2.1. Genus field k* with 3class number h = 3
and unit group index (U:V) = 9: The principal factors of k are either { p'q, p'^{2}q^{2} } or { p'^{2}q, p'q^{2} }, principal factors of k' coincide with those of k [3,5] . G = G^{(3)}(0,0,0) in ZEF(3,3), the metabelian 3group of maximal class 2 [1,4] , order 27, and exponent 3. Therefore H and H' = C_{3}*C_{3} in ZEF(2,2) and the capitulation type of k and of k' is (0,0,0,0), i. e., the complete 3class group C_{3}*C_{3} of k, resp. k', becomes principal in all four intermediate fields of k_{1}  k, resp. k'_{1} = k_{1}  k'. The Hilbert 3class field tower terminates after one step [3] . 



The first example of C(k) = C(k') = C_{3}*C_{3}
with capitulation type (0,0,0,0) and Hilbert 3class field tower of 1 step is [5] : f = 657 = 3^{2}*73 with (9/73)_{3} = 1 and (73/9)_{3} = 1 and with coinciding principal factors { 3^{2}*73, 3*73^{2} } in k and k'. 

2.2. Genus field k* with 3class number h = 9
and unit group index (U:V) = 27: The principal factors of k are either { p', p'^{2} } or { q, q^{2} }, principal factors of k' coincide with those of k [3,5] . G = G^{(4)}(0,1,0) in ZEF(4,4), a metabelian 3group of maximal class 3 [1,4] and order 81. Thus, H and H' = G^{(3)}(1,0,0) in ZEF(3,3) and the capitulation type of k and of k' is (1,1,1,1), i. e., only the ambiguous 3class group C_{3} of k, resp. k', becomes principal in any of the four intermediate fields of k_{1}  k, resp. k'_{1}  k'. (This is a situation which cannot occur for quadratic base fields.) The Hilbert 3class field tower becomes stationary after two steps [3] . 



Dan (02/08/26):
The minimal occurrence of C(k) = C(k') = C_{3}*C_{3} with capitulation type (1,1,1,1) and Hilbert 3class field tower of 2 steps is [5] : f = 2439 = 3^{2}*271 with (9/271)_{3} = 1 and (271/9)_{3} = 1 and with coinciding principal factors { 271, 271^{2} } in k and k'. 

Dan (02/09/11):
It should be pointed out that the famous example of A. Scholz and O. Taussky, which was mentioned several times in their papers, is exactly of this type [5] : f = 21451 = 19*1129 with (19/1129)_{3} = 1 and (1129/19)_{3} = 1 and with coinciding principal factors { 19, 19^{2} } in k and k'. 
References:
[1] Brigitte Nebelung, Klassifikation metabelscher 3Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989 [2] Charles J. Parry, Bicyclic Bicubic Fields, Canad. J. Math. 42 (1990), no. 3, 491  507 [3] Mohammed Ayadi, Sur la capitulation des 3classes d'idéaux d'un corps cubique cyclique, Thèse de doctorat, Université Laval, Québec, 1995 [4] Aïssa Derhem, Sur les corps cubiques cycliques de conducteur divisible par deux premiers, Casablanca, 2002 [5] Daniel C. Mayer, Class Numbers and Principal Factorizations of Families of Cyclic Cubic Fields with Discriminant d < 10^{10}, Univ. Graz, Computer Centre, 2002 
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