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Group theoretic approach to cyclic cubic fields

Siham Aouissi, Daniel C. Mayer

Abstract

Let (k1,k2,k3,k4) 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 k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary bicyclic, the automorphism group M = Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k 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 k1,k2,k3,k4. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G = Gal(F(3,infinity,k)/k) of the maximal unramified pro-3-extension of k.

Group theoretic approach to cyclic cubic fields

Abstract

Let (k1,k2,k3,k4) 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 k of the quartet whose 3-class group Cl(3,k) = Z/3Z x Z/3Z is elementary bicyclic, the automorphism group M = Gal(F(3,2,k)/k) of the maximal metabelian unramified 3-extension of k 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 k1,k2,k3,k4. With the aid of the relation rank d2(M), it is decided whether M coincides with the Galois group G = Gal(F(3,infinity,k)/k) of the maximal unramified pro-3-extension of k.
Paper Structure (26 sections, 54 theorems, 81 equations, 21 tables)

This paper contains 26 sections, 54 theorems, 81 equations, 21 tables.

Table of Contents

  1. Introduction
  2. Construction of cyclic fields of odd prime degree
  3. Multiplicity of conductors and discriminants
  4. Construction as ray class fields
  5. Arithmetic of cyclic cubic fields
  6. Rank of 3-class groups
  7. Ambiguous principal ideals
  8. Unramified extensions of cyclic cubic fields
  9. The absolute 3-genus field
  10. Capitulation kernels
  11. Capitulation targets
  12. Finite 3-groups of type (3,3)
  13. Categories I and II
  14. Category I, Graph 1
  15. Category I, Graph 2
  16. ...and 11 more sections

Key Result

Theorem 1

The conductor of a cyclic field of odd prime degree $\ell$ has the shape $c=\ell^e\cdot q_1\cdots q_\tau$, where $e\in\lbrace 0,2\rbrace$ and the $q_i$ are pairwise distinct prime numbers $q_i\equiv +1\,(\mathrm{mod}\,\ell)$, for $1\le i\le\tau$. The discriminant of $k=k_c$ is the perfect $(\ell-1)$ In the last case, we formally put $q_{\tau+1}:=\ell^2$. The number of non-isomorphic cyclic number

Theorems & Definitions (126)

  • Theorem 1
  • proof
  • Definition 1
  • Theorem 2
  • proof
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • ...and 116 more