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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. 7
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 (list; graph; refs; listen; history; text; internal format)



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.


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.


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.


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).


(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;


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

Sequence in context: A119711 A062589 A250235 * A094612 A250237 A112847

Adjacent sequences:  A250233 A250234 A250235 * A250237 A250238 A250239




Daniel Constantin Mayer, Nov 15 2014



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Last modified December 19 03:45 EST 2018. Contains 318245 sequences. (Running on oeis4.)