Galois scaffolds for cyclic \(p^n\)-extensions in characteristic \(p\) (Q2081523)

Let $K$ be a field of characteristic $p>0$. \textit{E. Witt} [J. Reine Angew. Math. 176, 126--140 (1936; JFM 62.1112.03)] showed that cyclic extensions $L/K$ can be described using the ring $W_{n}(K)$. The elements of $W_{n}(K)$ are indeed vectors with \(n\) entries taken from $K$, with nonstandard operations $\oplus$ and $\otimes$ which make $W_{n}(K)$ a commutative ring with 1. Witt showed that for a cyclic extension $L/K$ of degree $p^n$ there exists a vector $\beta\in W_{n}(K)$ such that $L$ is generated over $K$ by the coordinates of any solution in $W_{n}(K^{\mathrm{sep}})$ to the solution $\phi(x)=x\oplus\beta where \phi: W_{n}(K^{\mathrm{sep}})\to W_{n}(K^{\mathrm{sep}})$ is the map induced by the $p$-Frobenius on $K^{\mathrm{sep}}$; separable closure of $K$ [\textit{M. Demazure}, Lectures on \(p\)-divisible groups. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0247.14010)]. The authors of the paper under review use the previous joint work of the first author on totally ramified Galois extensions $L/K$ of degree $p^2$, [\textit{N. P. Byott} and \textit{G. G. Elder}, J. Number Theory 133, No. 11, 3598--3610 (2013; Zbl 1295.11133)] to generalize that results to give sufficient conditions on $\beta\in W_{n}(K)$ for the $C_{p^n}$-extension $L/K$ generated by the roots of $\phi(x)=x\oplus\beta$ to admit a Galois scaffold. The paper is a good source for interested researchers in the field.