Memorial 2009



Dedicated to the Memory of

Alexander Aigner

in the Year 2009:

Quadruplets of Totally Real Cubic Number Fields

Section 1. All totally real cubic fields L with discriminant 0 < d < 100000 and multiplicity m = 4

We give a coarse classification by means of 3-class numbers of the quadruplet,
and a fine classification distinguishing capitulation types a.2: (1,0,0,0) and a.3: (0,1,0,0).

Particularly interesting are the cases with one or more occurrences of 3-class number 9 (red color),
since they mark exceptions from the immense number of ground states of types a.2 and a.3.

This is merely the begin of the series of unramified quadruplets and has been
computed by Heider and Schmithals in 1982 on a CDC Cyber 76 at the University of Cologne [1].
Unfortunately, just the essential class number 9, which is characteristic for type a.1: (0,0,0,0),
has inadvertently been published as 3 [1, p.24].

Continuation

Counter n Discriminant d Regulators R and class numbers h as pairs (R, h) Capitulation type
1 32009 (6.7, 3) (8.8, 3) (14.1, 3) (45.1, 3) a.3: (0,0,0,3)
2 42817 (7.1, 3) (8.6, 3) (20.3, 3) (43.1, 3) a.3: (0,0,0,3)
3 62501 (9.1, 9) (23.6, 3) (25.4, 3) (38.2, 3) a.1: (0,0,0,0)
4 72329 (7.8, 6) (9.9, 3) (19.3, 3) (69.0, 3) a.2: (0,2,0,0)
5 94636 (23.4, 3) (32.8, 3) (51.7, 3) (80.0, 3) a.2: (0,0,3,0)


References:

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

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