Memorial 2009

Dedicated to the Memory of

Alexander Aigner

in the Year 2009:

Quadruplets of Totally Real Cubic Number Fields


The present contribution to the centennial festival with title "Memorial 2009"
in honour of my invaluable academic advisor, the number theorist Alexander Aigner,
is devoted to the fascinating realm of unramified cyclic cubic extensions N|K
of real quadratic number fields K with
elementary abelian bicyclic 3-class group Cl3(K) of type (3,3)
(i. e., isomorphic with the direct product of two cyclic groups C3 of order 3),
and the numerous interesting arithmetical phenomena arising in this context.

According to the combination of Artin's reciprocity law with the Galois correspondence,
these real quadratic number fields K have exactly four non-isomorphic unramified cyclic cubic extensions
N1, N2, N3, N4
sharing the same discriminant d3, where d denotes the discriminant of the maximal order of K.
This simple relation between the discriminants is due to the relative conductor f = 1 of unramified extensions.

Each of these unramified cyclic cubic relative extensions Ni|K with relative group Gal(Ni|K)=C3
is an absolute Galois extension with dihedral group Gal(Ni|Q)=D(6) of order 6 over the rational number field Q.
Therefore, by Galois theory, Ni contains three conjugate (and thus isomorphic) non-Galois absolutely cubic fields,
which we shall identify and designate by Li, since all their arithmetical invariants coincide.

By a quadruplet of totally real cubic number fields we understand
the family of these four non-isomorphic representatives of isomorphism classes
L1, L2, L3, L4,
all of whose members share the same fundamental discriminant d (equal to the quadratic discriminant),
and which is in one-to-one correspondence with the quadruplet
N1, N2, N3, N4.

We consider three closely related arithmetical problems for these quadruplets, the determination of
  • the capitulation (or principalization) type of K in the extensions Ni,
  • the structure of the group G=Gal(K2|K) of the second Hilbert 3-class field K2 of K,
  • the structure of the 3-class group Cl3(K1)=G'=Gal(K2|K1) of the first Hilbert 3-class field K1 of K.
For the solution of the first problem, fundamental systems of units are needed for K and all Li.
Here we mention that a complete, 2-dimensional capitulation is equivalent with a principal factorization type Alpha1,
whereas a partial, 1-dimensional capitulation is equivalent with a principal factorization type Delta1.
Type Alpha1 can be identified by the connection with the cohomology of the units,
but for the identification of type Delta1 the capitulation kernel must be determined.
The second and third problem can be solved by adequate theoretical statements, which we have developed in [7].

The earlier investigators Heider and Schmithals [2] (1982) used the table of Angell [1] to find the discriminants and units.
For our first extension [4] in August 1991, which was mainly devoted to the principal factorization types of ramified fields,
we computed the units with the aid of Voronoi's algorithm.
15 years later we started our extensive Real Capitulation Project in January 2006.
In our second extension [5] (February 2006), we utilized the table of Ennola and Turunen [3] to find the discriminants and generating polynomials,
performed Tschirnhausen transformations to obtain trace-free polynomials, and executed Voronoi's algorithm to get units.
However, we desired an independent verification of and a break through beyond the range of Ennola and Turunen.
In April 2006, we therefore computed all totally real cubic fields L with fundamental discriminants 0 < d < 106,
i. e., only those whose normal fields N are unramified over their quadratic subfields K.
The quadruplets L1, L2, L3, L4 of non-isomorphic fields sharing the same discriminant among these totally real cubic fields
are in one-to-one correspondence with the 161 real quadratic fields K of 3-class rank 2.
Among the latter we have 149 with 3-class group of type (3,3) and 12 with 3-class group of type (9,3).
This extensive database is the source of information for our third extension [6], which is presented here.


0 < d < 100000
100000 < d < 200000
200000 < d < 300000
300000 < d < 400000
Reaching the Border of Ennola and Turunen's Domain:
400000 < d < 500000
Breaking Through Beyond Ennola and Turunen's Domain:
500000 < d < 600000
600000 < d < 700000
700000 < d < 800000
800000 < d < 900000
900000 < d < 1000000


[1] Ian O. Angell,
A table of totally real cubic fields,
Math. Comp. 30 (1976), no. 133, 184-187.

[2] Franz-Peter Heider und Bodo Schmithals,
Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen,
J. reine angew. Math. 336 (1982), 1 - 25.

[3] Veikko Ennola and Reino Turunen,
On totally real cubic fields,
Math. Comp. 44 (1985), no. 170, 495-518.

[4] Daniel C. Mayer,
List of discriminants dL < 200000 of totally real cubic fields L,
arranged according to their multiplicities m and conductors f,

1991, Dept. of Comp. Sci., Univ. of Manitoba

[5] Daniel C. Mayer,
3-Capitulation over Quadratic Fields
with Discriminant |d| < 3*105 and 3-Class Group of Type (3,3)
Univ. Graz, Computer Centre, 2006.

[6] 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.

[7] Daniel C. Mayer,
Two-Stage Towers of p-Class Fields
over Base Fields with p-Class Group of Type (p,p)
(Latest Update)
Univ. Graz, 2009.

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