The capitulation problem in certain pure cubic fields

Siham Aouissi

doi:10.1007/s40316-025-00248-9

The capitulation problem in certain pure cubic fields

2025-09-01

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DOI：10.1007/s40316-025-00248-9

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Let F = Q((3)root n) be a pure cubic field with normal closure k = Q((3)root n, ), where n > 1 denotes a cube free integer, and & varsigma; is a primitive cube root of unity. Suppose k possesses an elementary bicyclic 3-class group Cl-3(k), and the conductor of k/Q(& varsigma;) has the shape f is an element of {pq(1)q(2), 3 pq, 9 pq} where p-1 (mod 9) and q, q(1), q(2)-2,5 (mod 9) are primes. It is disproved that there are only two possible capitulation types x(k), either type a.1, (0000), or type a.2, (1000). Evidence is provided, theoretically and experimentally, of two further types, b.10, (0320), and d.23, (1320).

Keywords:

Pure cubic number fields

Normal closure

Relative conductor

Multiplets

Cubic residue symbols

Differential principal factors

Unramified cyclic cubic extensions

Capitulation

3-Class field tower

Finite 3-groups

Elementary bicyclic commutator quotient

Maximal subgroups

Abelian quotient invariants

Kernels and targets of Artin transfers

The capitulation problem in certain pure cubic fields

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