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A Group Theoretic Approach to Cyclic Cubic Fields

Author

Listed:
  • 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)

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 .

Suggested Citation

  • Siham Aouissi & Daniel C. Mayer, 2023. "A Group Theoretic Approach to Cyclic Cubic Fields," Mathematics, MDPI, vol. 12(1), pages 1-53, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:126-:d:1310537
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