OFFSET
1,1
COMMENTS
This is the beginning of an investigation of the maximal unramified pro-3 extension of complex bicyclic biquadratic fields containing the third roots of unity which have an elementary 3-class group of rank two.
For the discriminants d in A250237, the 3-class field tower of K=Q(sqrt(-3),sqrt(d)) is abelian, terminating with the first stage at the Hilbert 3-class field already. An equivalent condition is that the second 3-class group G of K is given by G=SmallGroup(9,2). Another equivalent condition in terms of a fundamental system of units has been given by Yoshida.
REFERENCES
H. U. Besche, B. Eick, and E. A. O'Brien, The SmallGroups Library - a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA.
LINKS
D. C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2) (2012), 471-505.
E. Yoshida, On the 3-class field tower of some biquadratic fields, Acta Arith. 107 (2003), no. 4, 327-336.
EXAMPLE
PROG
(Magma)SetClassGroupBounds("GRH"); for n := 229 to 3000 do cnd := false; if (1 eq n mod 4) and IsSquarefree(n) then cnd := true; end if; if (0 eq n mod 4) then r := n div 4; if IsSquarefree(r) and ((2 eq r mod 4) or (3 eq r mod 4)) then cnd := true; end if; end if; if (true eq cnd) then R := QuadraticField(n); E := QuadraticField(-3); K := Compositum(R, E); C, mC := ClassGroup(K); if ([3, 3] eq pPrimaryInvariants(C, 3)) then s := Subgroups(C: Quot := [3]); a := [AbelianExtension(Inverse(mq)*mC) where _, mq := quo<C|x`subgroup> : x in s]; b := [NumberField(x) : x in a]; d := [MaximalOrder(x) : x in a]; b := [AbsoluteField(x) : x in b]; c := [MaximalOrder(x) : x in b]; c := [OptimizedRepresentation(x) : x in b]; b := [NumberField(DefiningPolynomial(x)) : x in c]; a := [Simplify(LLL(MaximalOrder(x))) : x in b]; if IsNormal(b[2]) then H := Compositum(NumberField(a[1]), NumberField(a[2])); else H := Compositum(NumberField(a[1]), NumberField(a[3])); end if; O := MaximalOrder(H); CH := ClassGroup(LLL(O)); if (0 eq #pPrimaryInvariants(CH, 3)) then n, ", "; end if; end if; end if; end for;
CROSSREFS
KEYWORD
hard,nonn
AUTHOR
Daniel Constantin Mayer, Nov 15 2014
STATUS
approved