The Capitulation Problem in Certain Pure Cubic Fields

Abstract: Let (\Gamma=\mathbb{Q}(\sqrt[3]{n})) be a pure cubic field with normal closure (k=\mathbb{Q}(\sqrt[3]{n},\zeta)), where (n>1) denotes a cube free integer, and (\zeta) is a primitive cube root of unity. Suppose (k) possesses an elementary bicyclic (3)-class group (\mathrm{Cl}_3(k)), and the conductor of (k/\mathbb{Q}(\zeta)) has the shape (f\in\lbrace pq_1q_2,3pq,9pq\rbrace) where (p\equiv 1\,(\mathrm{mod}\,9)) and (q,q_1,q_2\equiv 2,5\,(\mathrm{mod}\,9)) are primes. It is disproved that there are only two possible capitulation types (\varkappa(k)), either type (\mathrm{a}.1), ((0000)), or type (\mathrm{a}.2), ((1000)). Evidence is provided, theoretically and experimentally, of two further types, (\mathrm{b}.10), ((0320)), and (\mathrm{d}.23), ((1320)).