2022 年 76 巻 1 号 p. 101-118
Let k = ℚ(3√d, ζ3), where d > 1 is a cube-free positive integer, k0 = ℚ(ζ3) be the cyclotomic field containing a primitive cube root of unity ζ3, and G ＝ Gal(k / k0). The possible prime factorizations of d in our main result in previous work (Theorem 1.1 in Aouissi et al, Preprint, arXiv:1808.04678v2) give rise to new phenomena concerning the chain Θ = (θi)i∈ℤ of lattice minima in the underlying pure cubic subfield L = ℚ(3√d) of k. The aims of the present work are to give criteria for the occurrence of generators of primitive ambiguous principal ideals (ν) ∈ PkG / Pk0 among the lattice minima Θ = (θi)i∈ℤ of the underlying pure cubic field L = ℚ(3√d), and to explain the exceptional behavior of the chain Θ for certain radicands d with impact on determining the principal factorization type of L and k by means of Voronoi's algorithm.
