A Group Theoretic Approach to Cyclic Cubic Fields

Siham Aouissi and Daniel C. Mayer ()
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Siham Aouissi: Algebraic Theories and Applications Research Team (ATA), Ecole Normale Supèrieure of Moulay Ismail University (ENS-UMI), ENS, Toulal, Meknès B.P. 3104, Morocco
Daniel C. Mayer: Independent Researcher, Naglergasse 53, 8010 Graz, Austria

Mathematics, 2023, vol. 12, issue 1, 1-52

Abstract: Let ( k μ ) μ = 1 4 be a quartet of cyclic cubic number fields sharing a common conductor c = p q r divisible by exactly three prime(power)s, p , q , r . For those components of the quartet whose 3-class group Cl 3 ( k μ ) ≃ ( Z / 3 Z ) 2 is elementary bicyclic, the automorphism group M = Gal ( F 3 2 ( k μ ) / k μ ) of the maximal metabelian unramified 3-extension of k μ is determined by conditions for cubic residue symbols between p , q , r and for ambiguous principal ideals in subfields of the common absolute 3-genus field k * of all k μ . With the aid of the relation rank d 2 ( M ) , it is decided whether M coincides with the Galois group G = Gal ( F 3 ∞ ( k μ ) / k μ ) of the maximal unramified pro-3-extension of k μ .

Keywords: cyclic cubic number fields; conductor; combined cubic residue symbol; principal factors; absolute genus field; bicyclic bicubic fields; unramified cyclic cubic relative extensions; capitulation; maximal unramified pro-3-extension; finite 3-groups; elementary bicyclic commutator quotient; maximal subgroups; kernels of Artin transfers; abelian quotient invariants; relation rank (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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