OFFSET
1,1
COMMENTS
Generally, the 3-class ranks s of the real quadratic field R=Q(sqrt(d)) and r of the complex quadratic field C=Q(sqrt(-3d)) are related by the inequalities s <= r <= s+1. This reflection theorem was proved by Scholz and independently by Reichardt using a combination of class field theory and Kummer theory over the bicyclic biquadratic compositum K=R*E of R with Eisenstein's cyclotomic field E=Q(sqrt(-3)) of third roots of unity.
In particular, the biquadratic field K=Q(sqrt(-3),sqrt(d)) has a 3-class group of type (3,3) if and only if s=r and R and C both have 3-class groups of type (3).
Therefore, the discriminants in the sequence A250236 uniquely characterize all complex biquadratic fields containing the third roots of unity which have an elementary 3-class group of rank two.
The discriminant of K=R*E is given by d(K)=3^2*d^2 if gcd(3,d)=1 and simply by d(K)=d^2 if 3 divides d.
REFERENCES
G. Eisenstein, Beweis des Reciprocitätssatzes für die cubischen Reste in der Theorie der aus den dritten Wurzeln der Einheit zusammengesetzten Zahlen, J. Reine Angew. Math. 27 (1844), 289-310.
LINKS
H. Reichardt, Arithmetische Theorie der kubischen Körper als Radikalkörper, Monatsh. Math. Phys. 40 (1933), 323-350.
A. Scholz, Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. Reine Angew. Math. 166 (1932), 201-203.
EXAMPLE
A250236 is a proper subsequence of A250235. For instance, it does not contain the discriminant d=733, resp. 1373, although the corresponding real quadratic field R=Q(sqrt(d)) has 3-class group (3). The reason is that the 3-dual complex quadratic field C=Q(sqrt(-3d)) of R has 3-class group (9), resp. (27).
PROG
(Magma) for d := 2 to 3000 do a := false; if (1 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 (3 eq r mod 4)) then a := true; end if; end if; if (true eq a) then R := QuadraticField(d); E := QuadraticField(-3); K := Compositum(R, E); C := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;
CROSSREFS
KEYWORD
nonn
AUTHOR
Daniel Constantin Mayer, Nov 15 2014
STATUS
approved