A remark on higher circular \(\ell\)-units (Q1185168)

Let \(\ell\) be a prime number, and denote by \(\mu_{\ell^ \infty}\) the group of \(\ell^ n\)-th roots of unity, where \(n\) runs over all non- negative integers. In two earlier papers, the author and \textit{G. W. Anderson} had introduced and studied the group \(E_ \ell\) of higher circular \(\ell\)-units [cf. Ann. Math., II. Ser. 128, 271-293 (1988; Zbl 0692.14018) and Int. J. Math. 1, No. 2, 119-148 (1990; Zbl 0715.14021)]. The elements of \(E_ \ell\) appear as \(\ell\)-units in the maximal pro- \(\ell\) extension of \(\mathbb{Q}(\mu_{\ell^ \infty})\) ramified outside \(\ell\), and the Galois group \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) acts on \(E_ \ell\) via its canonical representation in the outer automorphism group of the pro-\(\ell\) fundamental group of \(\mathbb{P}^ 1- \{0,1,\infty\}\). The present brief note gives some additional information about the structure of the group \(E_ \ell\). The main result is the statement and the proof of the following theorem. Theorem: For any higher circular \(\ell\)-unit \(\varepsilon\in E_ \ell\) and any element \(\sigma\in\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\), \(\varepsilon^{\sigma-1}\) is a unit. This theorem says that if \(k\) is any finite Galois extension over \(\mathbb{Q}\) containing a higher circular \(\ell\)-unit \(\varepsilon\in E_ \ell\), then the fractional ideal \(\varepsilon O_ k\) over the ring \(O_ k\) of integers of \(k\) is \(\text{Gal}(k/\mathbb{Q})\)-invariant.