The capitulation problem in certain pure cubic fields

Abstract

Let \(\Gamma ={\mathbb {Q}}(\root 3 \of {n})\) be a pure cubic field with normal closure \(k={\mathbb {Q}}(\root 3 \of {n},\zeta ),\) 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 \({\textrm{Cl}}_3(k),\) and the conductor of \(k/{\mathbb {Q}}(\zeta )\) has the shape \(f\in \lbrace pq_1q_2,3pq,9pq\rbrace \) where \(p\equiv 1\,({\textrm{mod}}\,9)\) and \(q,q_1,q_2\equiv 2,5\,({\textrm{mod}}\,9)\) are primes. It is disproved that there are only two possible capitulation types \(\varkappa (k),\) either type \({\textrm{a}}.1,\) (0000),  or type \({\textrm{a}}.2,\) (1000). Evidence is provided, theoretically and experimentally, of two further types, \({\text {b}}.10,\) (0320),  and \({\textrm{d}}.23,\) (1320).