This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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 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 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.