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A250235 Discriminants of real quadratic fields with cyclic 3-class group (3). 8
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..55.

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.

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

A094612 is a subsequence, A006832 is a supersequence.

Sequence in context: A140017 A119711 A062589 * A250236 A094612 A250237

Adjacent sequences:  A250232 A250233 A250234 * A250236 A250237 A250238

KEYWORD

easy,nonn

AUTHOR

Daniel Constantin Mayer, Nov 14 2014

STATUS

approved

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Last modified December 16 01:36 EST 2017. Contains 296063 sequences.