OFFSET
1,1
COMMENTS
These real quadratic fields have class number divisible by 3 but not divisible by 9. Therefore, this sequence does not contain the discriminant 1129, since the corresponding quadratic field has cyclic 3-class group (9). However, this sequence contains the discriminant 697 whose corresponding quadratic field has class number 6=2*3. Note that 697 is not a member of the sequence A094612, where an exact class number 3 is required.
According to the Artin reciprocity law of class field theory, these real quadratic fields possess a cyclic cubic Hilbert 3-class field as their maximal unramified abelian 3-extension.
According to the Hasse formula d(K)=f^2*d for the discriminant d(K) of a non-Galois totally real cubic field in terms of the conductor f and the associated discriminant d of the real quadratic subfield of the normal closure of K, the sequence A006832 contains all discriminants d of real quadratic fields with class number divisible by 3, since they give rise to a totally real cubic field with conductor f=1 and discriminant d(K)=f^2*d=d. In particular, A006832 contains A250235.
LINKS
Emil Artin, Beweis des allgemeinen Reziprozitätsgesetzes, Abh. math. Sem. Univ. Hamburg 5 (1927), 353-363.
Helmut Hasse, Arithmetische Theorie der kubischen Körper auf klassenkörpertheoretischer Grundlage, Math. Z. 31 (1930), 565-582.
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 K := QuadraticField(d); C := ClassGroup(K); if ([3] eq pPrimaryInvariants(C, 3)) then d, ", "; end if; end if; end for;
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Daniel Constantin Mayer, Nov 14 2014
STATUS
approved