On formulas for the index of the circular distributions (Q1003137)

Let \(\mu_{s}\) be the set of \(s\)th roots of unity and \(\xi_{s}\) be a primitive \(s\)th root of unity in a fixed algebraic closure \(\mathbb Q^{\text{alg}}\) of \(\mathbb Q\). Let \(\mu_{\infty } = \cup _{s\in \mathbb N}\,\mu_{s}\), \(\mu_{s}^{*} = \mu_{s}\setminus \{1\}\), and \(\mu_{ \infty }^{*} = \mu_{\infty }\setminus \{1\}\), where \(\mathbb N\) is the set of positive integers. Let \(\varepsilon \in \mu_{ \infty }^{*}\) and \(d\in \mathbb N\); a circular distribution is a Galois equivariant map \(f\) from \(\mu_{ \infty }^{*}\) to \(\mathbb Q^{\text{alg}}\) such that \(f\) satisfies product conditions, \(\Pi _{\xi^{d} = \varepsilon}f(\xi) = f(\varepsilon )\), and congruence conditions, for each prime \(l\) and \(s \in N\) with \((l,s) = 1\), \(f(\varepsilon \xi ) = f(\xi )\) modulo primes over \(l\) for all \(\varepsilon \in \mu_{l}^{*}\), \(\xi \in\mu_{s}^{*}\). For such \(f\), let \(P^{f}_{s}\) be the group generated over \(\mathbb Z[\text{Gal}(Q(\mu_{s})/Q)]\) by \(f(\xi )\), \(\xi \in\mu_{s}\) and \(C^{f}_{s} = P^{f}_{s}\cap U_{s}\) where \(U_{s}\) denotes the global units of \(\mathbb Q(\mu_{s})\). In this paper, the author gives formulas for the indices \([P_{s}: P^{f}_{s}]\) and \([C_{s}: C^{f}_{s}]\) of \(P^{f}_{s}\) and \(C^{f}_{s}\) inside the circular numbers \(P_{s}\) and the Sinnott's units \(C_{s}\) over \(\mathbb Q(\mu_{s})\).