### Abstract

A circular distribution is a Galois equivariant map ψ from the roots of unity μ_{∞} to an algebraic closure of ℚ such that ψ satisfies product conditions, ∏_{ζ}^{d} =ε ψ (ζ) = ψ(ε) for ε ∈ μ_{∞} and d ∈ ℕ, and congruence conditions for each prime number l and s ∈ ℕ with (l, s) = 1, ψ(ε ζ) ≡ ψ(ζ) modulo primes over l for all ε ∈ μ_{l}, ζ ∈ μ_{s}, where μ_{l} and μ _{s} denote respectively the sets of lth and sth roots of unity. For such ψ, let Pψ_{s} be the group generated over ℤ[Gal(ℚ(μ_{s})/ℚ)] by ψ(ζ), ζ ∈ μ_{s} and let Cψ_{s} be Pψ_{s} ∩ U_{s}, where U_{s} denotes the global units of ℚ(μ_{s}). We give formulas for the indices (P_{s}:P ^{ψ}_{s}) and (C_{s} : C^{ψ} _{s}) of P^{ψ}_{s} and C^{ψ}_{s} inside the circular numbers P_{s} and units C_{s} of Sinnott over ℚ(μ_{s}).

Original language | English |
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Pages (from-to) | 381-396 |

Number of pages | 16 |

Journal | Manuscripta Mathematica |

Volume | 127 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2008 Nov 1 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)