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).
Bibliographical noteFunding Information:
This work was supported by the SRC Program of Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (No. R11-2007-035-01001-0). This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2006-312-C00455).
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