On the realization and classification of cyclic extensions of polynomial algebras over the Steenrod algebra

Abstract:

Suppose ${{\mathbf {R}}^*}$ is an unstable algebra over the Steenrod algebra of the form ${{\mathbf {P}}^*}(\sqrt [k]{d})$, where ${{\mathbf {P}}^*}$ is a polynomial algebra over the Steenrod algebra. If ${{\mathbf {R}}^*}$ is integrally closed then ${{\mathbf {R}}^*} = P{(V)^{{G_\mathcal {X}}}}$, where $C \leqslant GL(V)$ is generated by pseudoreflections and ${G_\mathcal {X}} = \ker \{ \mathcal {X}:G \to {\mathbf {F}}_p^*\}$ is a character of degree $k$.