For the 16 principal factorization types of cubic fields L in the following table we denote by
r ... the unit rank of the quadratic subfield k of the normal field N
z ... the 3-exponent of the subgroup of torsion units of k
u ... the 3-exponent of the unit norm index ( Uk : N(UN) )
a ... the 3-exponent of the group of absolute principal factors ( PLG : PQ )
b ... the 3-exponent of the group of relative principal factors ( DN|k-Ik : Ik )
c ... the 3-exponent of the capitulation kernel ( PN * Ik : Pk )
where the operator * denotes the intersection
PF-Type ... the principal factorization type (red color emphasizes Arnie's monster)
| Nr. | Signature | Designation | r | z | r + z | u | 1 + u | AbsPF a | RelPF b | CapPF c | PF-Type |
|---|---|---|---|---|---|---|---|---|---|---|---|
| 1 | (3,0) | Cyclic | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | ZETA |
| 2 | (3,0) | Totally Real | 1 | 0 | 1 | 1 | 2 | 0 | 0 | 2 | ALPHA 1 |
| 3 | 1 |
2 |
0 | 1 | 1 | ALPHA 2 | |||||
| 4 | 1 |
2 |
0 | 2 | 0 | ALPHA 3 | |||||
| 5 | 1 |
2 |
1 | 0 | 1 | BETA 1 | |||||
| 6 | 1 |
2 |
1 | 1 | 0 | BETA 2 | |||||
| 7 | 1 |
2 |
2 | 0 | 0 | GAMMA | |||||
| 8 | 0 | 1 | 0 | 0 | 1 | DELTA 1 | |||||
| 9 | 0 |
1 |
0 | 1 | 0 | DELTA 2 | |||||
| 10 | 0 |
1 |
1 | 0 | 0 | EPSILON | |||||
| 11 | (1,1) | Pure | 0 | 1 | 1 | 1 | 2 | 1 | 1 | 0 | ALPHA |
| 12 | 1 |
2 |
2 | 0 | 0 | BETA | |||||
| 13 | 0 | 1 | 1 | 0 | 0 | GAMMA | |||||
| 14 | (1,1) | Complex | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | ALPHA 1 |
| 15 | 0 |
1 |
0 | 1 | 0 | ALPHA 2 | |||||
| 16 | 0 |
1 |
1 | 0 | 0 | BETA |