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** ( U_{k} : N(U_{N}) )

a ... the 3-exponent of the group of **absolute principal factors** ( P_{L}^{G} : P_{Q} )

b ... the 3-exponent of the group of **relative principal factors** ( D_{N|k}^{-}I_{k} : I_{k} )

c ... the 3-exponent of the **capitulation kernel** ( P_{N} * I_{k} : P_{k} )

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 |