OFFSET
1,1
COMMENTS
The length of the Hilbert 3-class field tower of a complex quadratic field is infinite for 3-class rank at least 3, and it is 1 for 3-class rank 1. In contrast, the length is at least 2 but unbounded for 3-class rank 2, whence this is the only unsolved interesting case.
The terms 3299, 4027 and 9748 have been discussed in detail by Scholz and Taussky. In a footnote they also mention 3896 with an erroneous claim.
REFERENCES
H. Koch, B. B. Venkov, Über den p-Klassenkörperturm eines imaginär-quadratischen Zahlkörpers, Astérisque 24-25 (1975), 57-67.
LINKS
C. McLeman, p-tower groups over quadratic imaginary number fields, arXiv:1008.3003 [math.NT], 2010; Ann. Sci. Math. Québec 32 (2008), no. 2, 199-209.
A. Scholz and O. Taussky, Die Hauptideale der kubischen Klassenkörper imaginär-quadratischer Zahlkörper, J. Reine Angew. Math. 171 (1934), 19-41. DOI:10.1515/crll.1934.171.19
EXAMPLE
For n=1,4, resp. n=2,3, the 3-class group is of type (3,9), resp. (3,3).
PROG
(Magma)
for d := 2 to 10^5 do a := false; if (3 eq d mod 4) and IsSquarefree(d) then a := true; end if; if (0 eq d mod 4) then r := d div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (1 eq r mod 4)) then a := true; end if; end if; if (true eq a) then K := QuadraticField(-d); C := ClassGroup(K); if (2 eq #pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Daniel Constantin Mayer, May 24 2014
STATUS
approved