Joint research 2002

of Karim Belabas, Aïssa Derhem, and Daniel C. Mayer

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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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E-mail addresses:
Karim.Belabas@math.u-psud.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 2-prime conductor f = p*q,
where p = 1(mod 3) or p = 32 and q = 1(mod 3)
and where we put p' = 3 if p = 32 and p' = p otherwise,
we study the Hilbert 3-class field tower,
k = k0 <= k1 <= k2 <= ... <= kn <= ...,
the absolute genus field k0 <= k* <= k1,
and the possible capitulation types,
when the 3-class group of k is of type C3 or C3*C3.

G denotes the absolute Galois group Gal( k*1 | Q )
of the Hilbert 3-class field k*1 of the genus field k* [4] and
G = G1 >= G2 >= ... >= Gi >= ...
the descending central series [1] of G with Gi+1 = [Gi,G].
In particular, G2 = [G,G] is the commutator subgroup of G.

The genus field k* is the compositum k* = kp*kq of kp and kq,
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', kp, and kq.

We put H = Gal( k*1 | k ) and H' = Gal( k*1 | k' ).

1. The fields k with 3-class rank rho = 1
and 3-class group C3,
for which p and q are not mutual cubic residues.

Here the genus field k* has 3-class 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'2q2 } or { p'2q, p'q2 }
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 = C3*C3 in ZEF(2,2),
the abelian 3-group (of maximal class 1) [1,4] of order 9, and exponent 3.

Since H and H' = C3,
the capitulation type of k and of k' is trivial,
i. e., the (full) 3-class groups C3 of k and k' become principal in k*,
as Hilbert's Theorem 94 asserts in this simple case.

The Hilbert 3-class field tower terminates after one step [3] .
k* = k*1 = k1 = k2
//\\
kpkk'kq
\\//
Q
1 = H2 = G2
//\\
HpHH'Hq
\\//
G
The smallest examples of C(k) = C(k') = C3 are [5] :
a) f = 63 = 32*7 with (9/7)3 < > 1 and (7/9)3 < > 1
and with distinct principal factors { 32*7, 3*72 } in k
and { 3*7, 32*72 } in k'.
b) f = 91 = 7*13 with (13/7)3 = 1 but (7/13)3 < > 1
and with coinciding principal factors { 13, 132 } in k and k'.

2. The fields k with 3-class rank rho = 2
and 3-class group C3*C3,
for which p and q are cubic residues with respect to each other.
2.1. Genus field k* with 3-class number h = 3
and unit group index (U:V) = 9:

The principal factors of k are either { p'q, p'2q2 } or { p'2q, p'q2 },
principal factors of k' coincide with those of k [3,5] .

G = G(3)(0,0,0) in ZEF(3,3),
the metabelian 3-group of maximal class 2 [1,4] , order 27, and exponent 3.

Therefore H and H' = C3*C3 in ZEF(2,2) and
the capitulation type of k and of k' is (0,0,0,0),
i. e., the complete 3-class group C3*C3 of k, resp. k', becomes principal
in all four intermediate fields of k1 | k, resp. k'1 = k1 | k'.

The Hilbert 3-class field tower terminates after one step [3] .
k*1 = k1 = k2
|
k*
//\\
kpkk'kq
\\//
Q
1 = H2 = G3
|
G2
//\\
HpHH'Hq
\\//
G
The first example of C(k) = C(k') = C3*C3
with capitulation type (0,0,0,0)
and Hilbert 3-class field tower of 1 step is [5] :
f = 657 = 32*73 with (9/73)3 = 1 and (73/9)3 = 1
and with coinciding principal factors { 32*73, 3*732 } in k and k'.
2.2. Genus field k* with 3-class number h = 9
and unit group index (U:V) = 27:

The principal factors of k are either { p', p'2 } or { q, q2 },
principal factors of k' coincide with those of k [3,5] .

G = G(4)(0,1,0) in ZEF(4,4),
a metabelian 3-group 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 3-class group C3 of k, resp. k', becomes principal
in any of the four intermediate fields of k1 | k, resp. k'1 | k'.
(This is a situation which cannot occur for quadratic base fields.)

The Hilbert 3-class field tower becomes stationary after two steps [3] .
k*1 = k2 = k3
|
k1
|
k*
//\\
kpkk'kq
\\//
Q
1 = G4
|
H2 = G3
|
G2
//\\
HpHH'Hq
\\//
G
Dan (02/08/26):
The minimal occurrence of C(k) = C(k') = C3*C3
with capitulation type (1,1,1,1)
and Hilbert 3-class field tower of 2 steps is [5] :
f = 2439 = 32*271 with (9/271)3 = 1 and (271/9)3 = 1
and with coinciding principal factors { 271, 2712 } 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, 192 } in k and k'.

References:

[1] Brigitte Nebelung,
Klassifikation metabelscher 3-Gruppen
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 3-classes 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 < 1010,
Univ. Graz, Computer Centre, 2002

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