A363717 Prime radicands p == 1 (mod 9) of pure cubic number fields of type Gamma.

541, 919, 1279, 1531, 2161, 2269, 3637, 6211, 6427, 7129, 7723, 7867, 7993, 8389, 8461, 9649, 9901, 10009, 11071, 13627, 15031, 15391, 15607, 15661, 15787, 16741, 17713, 17911, 17929

Offset: 1

Author

Daniel Constantin Mayer, Jun 17 2023

Keywords

• nonn
• more

Comments

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.

Examples

			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.
		

Links

• S. Aouissi, D. C. Mayer, M. C. Ismaili, M. Talbi, and A. Azizi, 3-rank of ambiguous class groups of cubic Kummer extensions, Period. Math. Hungar. (2020) Vol. 81, 250-274.
• Siham Aouissi and Daniel C. Mayer, Disproof of claims about pure cubic fields, arXiv:2501.01361 [math.NT], 2025. See pp. 10, 15.
• M. C. Ismaili and R. El Mesaoudi, Sur la capitulation des 3-classes d'idéaux de la clôture normale de certains corps cubiques purs, Ann. Sci. Math. Québec, 29(2005), no. 1, 49-72.

Crossrefs

Subsequence of A061237. Via API related to A363699.

Programs

Magma

p:=1; while (p lt 10^5) do p:=NextPrime(p); if (1 eq p mod 9) then ZX:=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;