Memorial 2009

Dedicated to the Memory of

Alexander Aigner

in the Year 2009:

Quadruplets of Totally Real Cubic Number Fields

All totally real cubic fields L with discriminant 0 < d < 106 and multiplicity m = 4

Summary of statistical results concerning the second 3-class group [1]

For 128 of 149, i. e. 86 percent, of the investigated real quadratic base fields K
with discriminant 0 < d < 106 and elementary abelian bicyclic 3-class group Syl3Cl(K) of type (3,3)
the tower of unramified 3-class fields K < K1 < K2 ≤ … terminates at the second stage with K2.
More explicitly, these are the fields with capitulation types
a.2 (ground state), a.3*, a.3 (ground state), D.5, D.10.

The second 3-class group G=Gal(K2|K) is a metabelian 3-group of
order |G| = 3n and class cl(G) = m-1 with an isomorphism invariant e = n-m+2.
e-1 is the number of all bicyclic factors Gi/Gi+1 in the descending central series of G.
Among the 149 real quadratic fields K investigated, the second 3-class group G is of
  • maximal class, e=2, for the dominating part of 133 fields (89%) with capitulation types
    a.1 (0000) for 11 fields,
    a.2 (1000) for 34 fields,
    a.2 (1000) in the first excited state for a single field d=790085,
    a.3* (2000) for 34 fields,
    a.3 (2000) for 52 fields,
    a.3 (2000) in the first excited state for a single field d=494236,

  • second maximal class, e=3, for 15 fields (10%) with capitulation types
    c.18 (0313) for d=534824,
    c.21 (0231) for d=540365, 945813,
    D.5 (4224) for d=631769, 835853, 859064,
    D.10 (2241) for d=422573, 502796, 575729, 626411, 810661,
    E.9 (2231) for d=342664,
    G.19 (4321) for d=214712, 943077,
    H.4 (4443) for d=957013,

  • lower than second maximal class, e=4, for a single field with capitulation type
    b.10 (0043) for d=710652.

Unfortunately, no fields of capitulation types d.19, d.23, d.25 appeared,
since their occurrence is dependent on an associated absolute cubic field with 3-class number 27.
However, such a field has not been found up to now, and even 3-class number 9 is very rare.
In fact, among the 596 totally real cubic fields L involved, only 19 have 3-class number 9.
They occur for the capitulation types
a.1, a.2 (excited state), a.3 (excited state), c.18, c.21, b.10 (twice), and E.9.
Other cases with 3-class number 9 would be the missing capitulation types
d.19, d.23, d.25, E.6, E.8, E.14, F.7 (twice), F.11 (twice), F.12 (twice), F.13 (twice), and G.16.

The lack of the completely unknown capitulation types d.19, d.23, d.25
and some other types, known for complex quadratic base fields, E.6, E.8, E.14, F.7, F.11, F.12, F.13, G.16,
caused us to extend the Real Capitulation Project to the range 0 < d < 107 of discriminants.
This extension was enabled by our new Top Down Capitulation Algorithm,
based on the discovery of a connection between the structure of certain 3-class groups and capitulation types,
and would have been definitely impossible by using our implementation of the Algorithm of Scholz and Taussky,
since it is difficult to automatize the various parts of the computation.



[1] Daniel C. Mayer,
3-Capitulation over Quadratic Fields
with Discriminant |d| < 106 and 3-Class Group of Type (3,3)
(Latest Update)
Univ. Graz, Computer Centre, 2009.

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