%I #19 Sep 08 2022 08:46:10
%S 229,257,316,321,469,473,568,697,733,761,785,892,940,985,993,1016,
%T 1101,1229,1257,1304,1345,1373,1384,1436,1489,1509,1708,1765,1772,
%U 1901,1929,1937,1957,2021,2024,2089,2101,2177,2213,2233,2296,2429,2505,2557,2589,2636,2677,2713,2777,2857,2917,2920,2941,2981,2993
%N Discriminants of real quadratic fields with cyclic 3-class group (3).
%C 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.
%C 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.
%C 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.
%H Emil Artin, <a href="http://dx.doi.org/10.1007/BF02952531">Beweis des allgemeinen Reziprozitätsgesetzes</a>, Abh. math. Sem. Univ. Hamburg 5 (1927), 353-363.
%H Helmut Hasse, <a href="http://dx.doi.org/10.1007/BF01246435">Arithmetische Theorie der kubischen Körper auf klassenkörpertheoretischer Grundlage</a>, Math. Z. 31 (1930), 565-582.
%H Helmut Hasse, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002371952&IDDOC=85276">Arithmetische Theorie der kubischen Körper auf klassenkörpertheoretischer Grundlage</a>.
%o (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;
%Y A094612 is a subsequence, A006832 is a supersequence.
%K easy,nonn
%O 1,1
%A _Daniel Constantin Mayer_, Nov 14 2014