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 \(\mathrm{d}.23\), \((1320)\).