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%I #27 Jul 13 2023 02:50:49
%S 541,919,1279,1531,2161,2269,3637,6211,6427,7129,7723,7867,7993,8389,
%T 8461,9649,9901,10009,11071,13627,15031,15391,15607,15661,15787,16741,
%U 17713,17911,17929
%N Prime radicands p == 1 (mod 9) of pure cubic number fields of type Gamma.
%C According to their ambiguous principal ideals (API), the normal closures of pure cubic number fields can be classified into three types (see Aouissi et al.). For each type, the generating radicals (cube roots) are API. For type Alpha, relative API exist additionally. For type Beta, absolute API exist additionally. For type Gamma, only the radicals are API, but some unit of the normal closure has a primitive third root of unity as its cyclic cubic relative norm. The latter property is characteristic for type Gamma. If the radicand is a prime p == 1 (mod 9) then type Beta is excluded, and, statistically, type Alpha dominates by far. The present subsequence of radicands with type Gamma is very sparse. Ismaili and El Mesaoudi have proved an important application of this subsequence. Type Gamma enables more capitulation types of closely related normal closures in their unramified cyclic cubic extensions than type Alpha.
%D M. C. Ismaili et R. El Mesaoudi, Sur la capitulation des 3-classes d'idéaux de la clôture normale de certains corps cubiques purs, Ann. Sci. Math. Quebec, 29(2005), no. 1, 49-72.
%H S. Aouissi et al., <a href="https://doi.org/10.1007/s10998-020-00326-1">3-rank of ambiguous class groups of cubic Kummer extensions</a>, Period. Math. Hungar., 81(2020), 250-274.
%e The initial term 541 is the 16th term of the sequence of primes p == 1 (mod 9). The closely related conductors c = 3*2*541 = 3246 and c = 3*5*541 = 8115, resp. c = 9*541 = 4869, give rise to rare capitulation types d.23, (1320), resp. a.2, (1000), which cannot occur for type Alpha, according to Ismaili and El Mesaoudi.
%o (Magma)
%o p:=1; while (p lt 10^5) do p:=NextPrime(p); if (1 eq p mod 9) then ZX<X>:=PolynomialRing(Integers()); L:=NumberField(X^3-p); K:=NumberField(X^2+X+1); N:=Compositum(L,K); SetClassGroupBounds("GRH"); CL:=ClassGroup(L); VL:=Valuation(#CL,3); CN:=ClassGroup(N); VN:=Valuation(#CN,3); E:=VN-2*VL+1; if (1 eq E) then printf "%o, ",p; end if; end if; end while;
%Y Subsequence of A061237. Via API related to A363699.
%K nonn,more
%O 1,1
%A _Daniel Constantin Mayer_, Jun 17 2023
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