A generalization of Kummer theory to Hopf-Galois extensions (Q6612135)

Let \(n\) be any natural number. Let \(K\) be a field whose characteristic is \(0\) or coprime to \(n\). In the case that \(K\) contains a primitive \(n\)-th root \(\zeta_n\) of unity, a finite extension \(L/K\) is said to be \textit{Kummer with respect to \(n\)} if \(L/K\) is a Galois extension whose Galois group is abelian of exponent \(n\). It is well-known that the Kummer extensions of \(K\) with respect to \(n\) are exactly those of the form\N\[\NL = K(\alpha_1,\dots,\alpha_k),\N\]\Nwhere \(\alpha_1^n,\dots,\alpha_k^n\in K\) and \(n\) is minimal with respect to this last condition. Here, it is crucial that \(\zeta_n\in K\). The paper under review drops this assumption and developed a generalized Kummer theory via the use of Hopf-Galois structures.\N\NLet \(L/K\) be a finite extension and let \(H\) be a Hopf-Galois structure on it. As an analog of Kummer extensions, the author defines \(L/K\) to be \textit{\(H\)-Kummer} if \(L=K(\alpha_1,\dots,\alpha_k)\) for some \(\alpha_1,\dots,\alpha_k\in L\) that are all \(H\)-eigenvectors, namely, eigenvectors of \(h\) for all \(h\in H\).\N\NBy considering almost classically Galois structures, the author was able to translate the correspondence between Kummer extensions and radical extensions to a more general situation.\N\NRecall that a finite separable extension \(L/K\) with Galois closure \(\widetilde{L}\) is said to be \textit{almost classically Galois} if \(\mathrm{Gal}(\widetilde{L}/L)\) has a normal complement, \(J\) say, in the group \(\mathrm{Gal}(\widetilde{L}/K)\). In this case, the fixed field \(M\) of \(J\) is called a \textit{complement} of \(L/K\). It is known that \(M\) can be used to construct a Hopf-Galois structure \(H_M\) on \(L/K\), called the \textit{almost classically Galois structure corresponding to \(M\)}, and the Hopf--Galois correspondence associated to \(H_M\) is bijective [\textit{C. Greither} and \textit{B. Pareigis}, J. Algebra 106, 239--258 (1987; Zbl 0615.12026)].\N\NThe author proved the following result in Theorem 1.1. We refer the reader to the paper under review for the undefined terminology.\N\N\textbf{Theorem.} Let \(K\) be a field whose characteristic is \(0\) or coprime to \(n\). Let \(M= K(\zeta_n)\) and let \(L/K\) be a strongly decomposable extension of the form\N\[\NL = K(\alpha_1,\dots,\alpha_k),\N\]\Nwhere \(\alpha_1,\dots,\alpha_k\in L\) are such that \(K(\alpha_i)\) and \(K(\alpha_j)\) are strongly disjoint almost classically Galois extensions whenever \(i\neq j\). Then the following are equivalent.\N\begin{itemize}\N\item[1.] \(L\cap M = K\), and \(\alpha_1^n,\dots,\alpha_k^n\in K\) with \(n\) minimal with respect to this property.\N\item[2.] \(M\) is a complement of \(L/K\), the extension \(\widetilde{L}/M\) is Kummer with respect to \(n\), and \(\alpha_1,\dots,\alpha_k\) are all \(H_M\)-eigenvectors.\N\end{itemize}\NNote that the extension \(L/K\) is \(H_M\)-Kummer in this case.\N\NThere have been various research that applies Hopf-Galois structures in the study of Galois module theory (see [\textit{L. N. Childs}, Taming wild extensions: Hopf algebras and local Galois module theory. Providence, RI: American Mathematical Society (AMS) (2000; Zbl 0944.11038)]). Continuing research in this direction, the author also investigated the Galois module structure of rings of integers in \(H\)-Kummer extensions and obtained the following result in Theorem 1.2.\N\N\textbf{Theorem.} Let \(K\) be the field of fractions of some Dedekind domain \(\mathcal{O}_K\). Let \(L/K\) be a finite extension and let \(\mathcal{O}_L\) denote the integral closure of \(\mathcal{O}_K\) in \(L\). Suppose that \(H\) is a Hopf--Galois structure on \(L/K\) such that \(L/K\) is \(H\)-Kummer and \(\mathcal{O}_L\) admits an \(\mathcal{O}_K\)-basis \(\{\gamma_j\}_{j=1}^{n}\) consisting of \(H\)-eigenvectors. Then \(\mathcal{O}_L\) is free over the associated order of \(H\) and any element of the form\N\[\N\beta = \sum_{j=1}^{n} \beta_j\gamma_j,\N\]\Nwhere \(\beta_1,\dots,\beta_n\in \mathcal{O}_K^\times\), is a generator.