The Capitulation Problem in Certain Pure Cubic Fields

TL;DR

This paper investigates capitulation types in certain pure cubic fields with specific class group structures and conductors, providing theoretical and experimental evidence for additional capitulation types beyond previously known classifications.

Contribution

It disproves the exclusivity of two capitulation types in these fields and introduces evidence for two new types, expanding the understanding of capitulation phenomena.

Findings

Disproved only two capitulation types exist in the studied fields.

Provided theoretical and experimental evidence for two additional capitulation types.

Enhanced classification of capitulation behavior in pure cubic fields with specific conductors.

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