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

Abstract

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