Galois module structure of some elementary \(p\)-abelian extensions (Q6584663)

Let \(p\) be any prime. The set-up is as follows:\N\begin{itemize}\N\item \(F\) is field for which the maximal pro-\(p\) quotient of the absolute Galois group of \(F\) is a free and finitely generated pro-\(p\) group;\N\item \(F\) is assumed to contain a primitive \(p\)th root of unity when \(\mathrm{char}(F)\neq p\);\N\item \(K\) is a Galois extension of \(F\) for which \(\mathrm{Gal}(K/F)\) is a finite \(p\)-group.\N\end{itemize}\NConsider the parameterizing space \(J(K)\) of elementary \(p\)-abelian extensions of \(K\). By Artin-Schreier theory and Kummer theory, respectively, it is known that\N\[\NJ(K) = \begin{cases} K/\wp(K) & \mbox{if }\mathrm{char}(K)=p,\\\NK^\times/K^{\times p} &\mbox{if }\mathrm{char}(K)\neq p, \end{cases}\N\]\Nwhere \(\wp(K) = \{k^p - k :k\in K\}\) in the former case. Clearly \(J(K)\) is equipped with a natural \(\mathbb{F}_p\)-vector space structure. Denote by\N\[\N[F] = \begin{cases} (F+\wp(K))/\wp(K) & \mbox{if }\mathrm{char}(K)=p,\\\NFK^{\times p}/K^{\times p} &\mbox{if }\mathrm{char}(K)\neq p, \end{cases}\N\]\Nthe subspace of \(J(K)\) consisting of the elements which have a representative from \(F\).\N\NSince \(K/F\) is a Galois extension in the current context, one can also view \(J(K)\) as an \(\mathbb{F}_p[\mathrm{Gal}(K/F)]\)-module. In the case that \(\mathrm{Gal}(K/F)\) is a cyclic \(p\)-group, the module structure of \(J(K)\) has already been computed. In the paper under review, the authors allow \(\mathrm{Gal}(K/F)\) to be any finite \(p\)-group, and they were able to prove that\N\[\NJ(K) \simeq \Omega^{-2}_{\mathbb{F}_p[\mathrm{Gal}(K/F)]}(\mathbb{F}_p)\oplus Y,\N\]\Nwhere \( \Omega_{\mathbb{F}_p[\mathrm{Gal}(K/F)]}\) denotes the Heller operator and \(Y\) is a free module of rank \(\dim[F]\).