### 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)

### Cite this

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*Manuscripta Mathematica*, vol. 127, no. 3, pp. 381-396. https://doi.org/10.1007/s00229-008-0214-7

**On formulas for the index of the circular distributions.** / Seo, Soogil.

Research output: Contribution to journal › Article

TY - JOUR

T1 - On formulas for the index of the circular distributions

AU - Seo, Soogil

PY - 2008/11/1

Y1 - 2008/11/1

N2 - 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, ∏ζd =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 ∈ μl, ζ ∈ μs, where μl and μ s denote respectively the sets of lth and sth roots of unity. For such ψ, let Pψs be the group generated over ℤ[Gal(ℚ(μs)/ℚ)] by ψ(ζ), ζ ∈ μs and let Cψs be Pψs ∩ Us, where Us denotes the global units of ℚ(μs). We give formulas for the indices (Ps:P ψs) and (Cs : Cψ s) of Pψs and Cψs inside the circular numbers Ps and units Cs of Sinnott over ℚ(μs).

AB - 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, ∏ζd =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 ∈ μl, ζ ∈ μs, where μl and μ s denote respectively the sets of lth and sth roots of unity. For such ψ, let Pψs be the group generated over ℤ[Gal(ℚ(μs)/ℚ)] by ψ(ζ), ζ ∈ μs and let Cψs be Pψs ∩ Us, where Us denotes the global units of ℚ(μs). We give formulas for the indices (Ps:P ψs) and (Cs : Cψ s) of Pψs and Cψs inside the circular numbers Ps and units Cs of Sinnott over ℚ(μs).

UR - http://www.scopus.com/inward/record.url?scp=53649107425&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=53649107425&partnerID=8YFLogxK

U2 - 10.1007/s00229-008-0214-7

DO - 10.1007/s00229-008-0214-7

M3 - Article

AN - SCOPUS:53649107425

VL - 127

SP - 381

EP - 396

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 3

ER -