A250235 Discriminants of real quadratic fields with cyclic 3-class group (3).

229, 257, 316, 321, 469, 473, 568, 697, 733, 761, 785, 892, 940, 985, 993, 1016, 1101, 1229, 1257, 1304, 1345, 1373, 1384, 1436, 1489, 1509, 1708, 1765, 1772, 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

Offset: 1

Author

Daniel Constantin Mayer, Nov 14 2014

Keywords

• easy
• nonn

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.
• Helmut Hasse, Arithmetische Theorie der kubischen Körper auf klassenkörpertheoretischer Grundlage.

Crossrefs

A094612 is a subsequence, A006832 is a supersequence.

Programs

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;