A363699 Radicands of pure cubic number fields of type BETA and subtype M0.

2, 455, 833, 850, 1078, 1235, 1430, 1573, 3857, 4901, 6061, 6358, 6370, 8294, 8959, 9922, 11284, 12121, 12673, 12818, 14801, 17986, 18241, 20539, 21607, 22747, 23218, 26474, 27115, 29716, 30073, 31046, 32062, 32269, 33337, 36518, 37570, 38399, 38657, 38686, 39146, 40223, 41990, 42143

Offset: 1

Author

Daniel Constantin Mayer, Jul 09 2023

Keywords

• nonn

Comments

According to their differential principal factors (DPF), the normal closures of pure cubic number fields can be classified into three types ALPHA, BETA, GAMMA (see Aouissi et al., Period. Math. Hungar.). For each type, the generating radicals (cube roots) are DPF. For type BETA, absolute DPF exist additionally. Type BETA can be subdivided further into three subtypes (see Aouissi et al., Kyushu J. Math.). For each subtype, the units form an orbit of lattice minima in the maximal order of the pure cubic field. For subtype M2, resp. M1, resp. M0, two, resp. one, resp. no, further orbit(s) of lattice minima, consisting of non-unital DPF with principal factor norms, exist additionally. The exotic subtype M0 has the fatal drawback that the Voronoi algorithm, which recursively constructs the chain of lattice minima, fails to detect non-unital DPF, although they exist, and thus is unable to find the correct classification into type BETA. While the coarse types ALPHA, BETA, GAMMA can be distinguished by means of MAGMA or PARI/GP, no modern computer algebra system possesses the required routines to resolve the fine subtypes M2, M1, and M0.

Examples

			Daniel Constantin Mayer discovered that two radicands of M0-fields, 1430 and 12673, both of Dedekind species II, D == 1,8 (mod 9), and three further radicands of M0-fields, 6370, 9922, 11284, all of Dedekind species IB, D == 2,4,5,7 (mod 9), are missing from the table by H. C. Williams, Math. Comp., Section 6, Table 2, p. 273.
		

References

• S. Aouissi, A. Azizi, M. C. Ismaili, D. C. Mayer, M. Talbi, Principal factors and lattice minima in cubic fields, Kyushu J. Math. 76 (2022), No. 1, 101-118.
• Daniel Constantin Mayer, Table of pure cubic number fields with normalized radicands between 0 and 110000, Karl-Franzens-Universität, Graz, April 1989.
• Daniel Constantin Mayer, The algorithm of Voronoi for orders in simply real cubic number fields, Karl-Franzens-Universität, Graz, March 1989.
• Daniel Constantin Mayer, Differential principal factors and units in pure cubic number fields, Karl-Franzens-Universität, Graz, August 1988.
• G. F. Voronoi, Ob odnom obobshchenii algorithma nepreryvnykh drobei (On a generalization of the algorithm of continued fractions). Doctoral Dissertation, Warsaw, 1896 (in Russian).

Links

• Daniel Constantin Mayer, Table of n, a(n) for n = 1..92
• S. Aouissi et al., 3-rank of ambiguous class groups of cubic Kummer extensions, Period. Math. Hungar., 81(2020), 250-274.
• Daniel Constantin Mayer, Detailed comments and examples
• Daniel Constantin Mayer, Magma program
• Daniel Constantin Mayer, Fast Voronoi Algorithm (Magma)
• H. C. Williams, Determination of principal factors in Q(D^1/2) and Q(D^1/3), Math. Comp. 38 (1982), No. 157, 261-274.

Crossrefs

Cf. A363717.