On the Galois module structure of cyclic Kummer extensions (Q958634)

Let \(K/k\) be a finite Galois extension of fields with Galois group \(G\). Then the Normal Basis Theorem says that \(K\) is a free rank \(1\) module over the group algebra \(k[G]\). Now suppose that \(K\) and \(k\) are either number fields or \(p\)-adic fields, and let \(\mathfrak{O}_{K}\) and \(\mathfrak{O}_{k}\) respectively denote their rings of integers. Let \(\mathfrak{A} := \{ \lambda \in k[G] \mid \lambda \mathfrak{O}_{K} \subseteq \mathfrak{O}_{K} \}\) be the associated order. Then a natural problem is to determine \(\mathfrak{A}\) and the structure of \(\mathfrak{O}_{K}\) as a (left) \(\mathfrak{A}\)-module. When \(K/k\) is (at most) tamely ramified, \(\mathfrak{O}_{K}\) is free (resp.\ locally free) over \(\mathfrak{A} = \mathfrak{O}_{k}[G] \) in the \(p\)-adic case (resp.\ number field case). However, the situation is somewhat more complicated when \(K/k\) is wildly ramified; in particular \(\mathfrak{O}_{k}[G] \subsetneq \mathfrak{A}\). The first part of the paper under review considers situation in which \(K/k\) is a totally and wildly ramified cyclic Kummer extension of \(p\)-adic fields of odd prime power degree \(p^{n}\). Suppose that \(K=k(\alpha)\) where \(\alpha^{p^{n}} \in \mathfrak{O}_{k}\) and \((\mathrm{val}_{K}(\alpha-1),p)=1\). In [\textit{Y.\ Miyata}, Math. Proc. Camb. Philos. Soc. 123, No. 2, 199--212 (1998; Zbl 1073.11525)], the author gave a necessary and sufficient condition for \(\mathfrak{O}_{K}\) to be free (necessarily of rank \(1\)) over \(\mathfrak{A}\). Note that every intermediate field of \(K/k\) is of the form \(K_{l} := k(\alpha^{p^{n-l}})\) with \(0 \leq l \leq n\). Let \(\mathfrak{O}_{l}\) denote the ring of integers of \(K_{l}\) and let \(\mathfrak{A}_{l}\) be the associated order of \(K_{l}/k\) contained in the group algebra \(k[\mathrm{Gal}(K_{l}/k)]\). Theorem 4 of the paper under review gives a condition for \(\mathfrak{O}_{l}\) to be free over \(\mathfrak{A}_{l}\) for \textit{every} \(l\). This condition involves the first ramification number \(c_{1}\) of \(K/k\) and is as follows: there exist \(m\) with \(0 \leq m < n\) and \(a_{0}\) dividing \(p-1\) such that \(c_{1} \equiv a_{0}(p^{m} + p^{m-1} + \cdots + 1) \mod p^{n}\). In Theorem 6, the author then goes on to give an explicit description of \(\mathfrak{A}\), which allows him to show that \(\mathfrak{A}\) is stable under the action of \(C:=\mathrm{Aut}(G) \cong (\mathbb{Z}/p^{n}\mathbb{Z})^{\times}\). Now let \(k\) be an arbitrary number field and \(G\) be a finite abelian group. Suppose that \(K/k\) is a tamely ramified Galois extension with \(\mathrm{Gal}(K/k) \cong G\). Then \(\mathfrak{O}_{K}\) determines a class \((\mathfrak{O}_{K})\) in the locally free class group \(\mathrm{Cl}(\mathfrak{O}_{k}[G])\). Let \(R(\mathfrak{O}_{k}[G])\) denote the subset of all such classes. In [\textit{L.\ R.\ McCulloh}, J. Reine Angew. Math. 375/376, 259--306 (1987; Zbl 0619.12008)], this set of ``realizable classes'' was determined and is shown to be a subgroup of \(\mathrm{Cl}(\mathfrak{O}_{k}[G])\). In the second part of the paper under review, the author adapts McCulloh's methods to consider the following situation. Let \(G\) be a cyclic group of odd prime power degree \(p^{n}\), let \(k\) be a number field containing a primitive \(p^{n}\)th root of unity and let \(\mathfrak{A}\) be a fixed order in \(k[G]\). Let \(R(\mathfrak{A})\) be the subset of \(\mathrm{Cl}(\mathfrak{A})\) of classes realized by rings of integers \(\mathfrak{O}_{K}\) of wildly ramified extensions \(K/k\), with \(\mathrm{Gal}(K/k) \cong G\), whose completions at the primes above \(p\) are of the form considered in the first part of the paper, and for which \(\mathfrak{O}_{K}\) is locally free over \(\mathfrak{A}\). Theorem 9 gives an adelic description of \(R(\mathfrak{A})\) as a coset of a certain subgroup in \(\mathrm{Cl}(\mathfrak{A})\). The final result concerns \(R(\mathfrak{A}[\mu_{E}])\), the image of \(R(\mathfrak{A})\) in \(\mathrm{Cl}(\mathfrak{A}[\mu_{E}])\), where \(\mathfrak{A}[\mu_{E}]\) is the quotient of \(\mathfrak{A}\) obtained by restricting to the faithful characters of \(G\). Theorem 11 shows that \(R(\mathfrak{A}[\mu_{E}])\) contains a certain coset of \(\mathrm{Cl}(\mathfrak{A}[\mu_{E}])^{S_{C}}\), where \(S_{C}\) is the Stickelberger ideal contained in \(\mathbb{Z}[C]\).