A Group Theoretic Approach to Cyclic Cubic Fields

Abstract

Let (kμ)μ=14 be a quartet of cyclic cubic number fields sharing a common conductor c=pqr divisible by exactly three prime(power)s, p,q,r. For those components of the quartet whose 3-class group Cl3(kμ)≃(Z/3Z)2 is elementary bicyclic, the automorphism group M=Gal(F32(kμ)/kμ) of the maximal metabelian unramified 3-extension of kμ 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 kμ. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G=Gal(F3∞(kμ)/kμ) of the maximal unramified pro-3-extension of kμ.

Keywords

• cyclic cubic number fields
• conductor
• combined cubic residue symbol
• principal factors
• absolute genus field
• bicyclic bicubic fields