Real quadratic fields with 3-class field tower of length 2.
Since the capitulation types a.2 and a.3 are associated with
2-stage metabelian 3-groups G with commutator factor group of type (3,3),
whose main commutator has [6] the symbolic order Y2 = (X2,Y),
we have many new examples of 3-class field towers of length 2
for each of the following discriminants:
1. 32009
2. 42817
4. 72329
5. 94636
6. 103809
7. 114889
8. 130397
9. 142097
10. 151141
12. 153949
13. 172252
14. 173944
15. 184137
16. 189237
17. 206776
18. 209765
19. 213913
20. 214028
22. 219461
23. 220217
24. 250748
26. 259653
27. 265245
28. 275881
29. 283673
30. 298849
This is due to the fact that
G2 = G´ = Z[X,Y]/Y2 is of type (3,3)
and thus the next members of the descending central series [5] are
G3 = (3) and
G4 = 1.
However, the condition
G4 = 1
warrants a 3-class field tower of length 2,
according to Scholz/Taussky [1], Heider/Schmithals [2], and Brink/Gold [3,4].
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Note that we cannot state any claim concerning
the length of the 3-class field tower over
the real quadratic fields with discriminants
3. 62501
11. 152949
21. 214712
25. 252977
37. 342664
since
Z[X,Y]/Z is of type (3,3,3,3)
for d = 214712
(where Z=(3,X3,XY,Y3,X2-Y2)
denotes the symbolic order of the irregular critical case),
Z[X,Y]/X4 is of type (9,9,3)
for d = 342664, and
Z[X,Y]/Y4 is of type (9,9)
for the remaining three cases
d = 62501, d = 152949, and d = 252977.
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However, we have also a new example of a
finite 3-class field tower of length 2 over
the real quadratic field with discriminant
45. 422573
because
G2 = G´ = Z[X,Y]/L2 is of type (3,3,3)
and the next members of the descending central series [5] are
G3 = (3,3) and
G4 = 1.
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