Capitulation Types of Pure Cubic Fields with Bicyclic 3-Class Group
The capitulation type of the normal closure of a pure cubic field records how the 3-part of its ideal class group becomes principal (capitulates) in each unramified cyclic cubic extension, encoded as the kernels of Artin transfer homomorphisms. For pure cubic fields whose sextic dihedral normal closure has an elementary bicyclic 3-class group and a conductor of prescribed shape, more capitulation types occur than a widely cited earlier classification allowed, and the flawed group-theoretic elimination argument behind that classification is identified and corrected.
THE CAPITULATION PROBLEM IN CERTAIN PURE CUBIC FIELDS SIHAM AOUISSI AND DANIEL C. MAYER Abstract.
For a pure cubic field Gamma = Q(n^{1/3}) (n>1 cubefree) with dihedral degree-6 normal closure k = Q(n^{1/3}, zeta), the capitulation type kappa(k) records, for each of the four unramified cyclic cub…