 # Finite 3-class field towers of length 2

 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  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  are G3 = (3) and G4 = 1. However, the condition G4 = 1 warrants a 3-class field tower of length 2, according to Scholz/Taussky , Heider/Schmithals , and Brink/Gold [3,4]. 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. 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  are G3 = (3,3) and G4 = 1.

 References:  Arnold Scholz und Olga Taussky, Die Hauptideale der kubischen Klassenkörper imaginär quadratischer Zahlkörper, J. reine angew. Math.171 (1934), 19 - 41.  Franz-Peter Heider und Bodo Schmithals, Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen, J. reine angew. Math. 336 (1982), 1 - 25.  James R. Brink, The class field tower for imaginary quadratic number fields of type (3,3), Dissertation, Ohio State Univ., 1984.  James R. Brink and Robert Gold, Class field towers of imaginary quadratic fields, manuscripta math. 57 (1987), 425 - 450  Brigitte Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3,3) und Anwendung auf das Kapitulationsproblem, Inauguraldissertation, Köln, 1989.  Daniel C. Mayer, Two-Stage Towers of 3-Class Fields over Quadratic Fields, Univ. Graz, 2006.