A250236 Fundamental discriminants d such that the real quadratic field Q(sqrt(d)) and the complex quadratic field Q(sqrt(-3d)) both have cyclic 3-class groups of order 3.

229, 257, 316, 321, 469, 473, 568, 697, 761, 785, 892, 940, 985, 993, 1016, 1229, 1304, 1345, 1384, 1436, 1489, 1509, 1708, 1765, 1929, 1937, 2024, 2089, 2101, 2177, 2233, 2296, 2505, 2557, 2589, 2677, 2920, 2941, 2981, 2993

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

Table of n, a(n) for n=1..40.

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

A250235 and A094612 are supersequences, A250237, A250238, A250239, A250240, A250241, A250242 are pairwise disjoint subsequences.

Sequence in context: A062589 A342368 A250235 * A094612 A250237 A350165

Adjacent sequences: A250233 A250234 A250235 * A250237 A250238 A250239

Keyword

nonn

Author

Daniel Constantin Mayer, Nov 15 2014

Status

approved