Inseparable extensions of algebras over the Steenrod algebra with applications to modular invariant theory of finite groups

We consider purely inseparable extensions $\textrm{H}\hookrightarrow \sqrt[\mathscr{P}^*]{\textrm{H}}$ of unstable Noetherian integral domains over the Steenrod algebra. It turns out that there exists a finite group $G\le\textrm{GL}(V)$ and a vector space decomposition $V=W_0\oplus W_1\oplus\dotsb\oplus W_e$ such that $\overline{\textrm{H}}=(\mathbb{F}[W_0] \otimes\mathbb{F}[W_1]^p\otimes\dotsb\otimes\mathbb{F}[W_e]^{p^e})^G$ and $\overline{\sqrt[\mathscr{P}^*]{\textrm{H}}}=\mathbb{F}[V]^G$, where $\overline{(-)}$ denotes the integral closure. Moreover, $\textrm{H}$ is Cohen-Macaulay if and only if $\sqrt[\mathscr{P}^*]{\textrm{H}}$ is Cohen-Macaulay. Furthermore, $\overline{\textrm{H}}$ is polynomial if and only if $\sqrt[\mathscr{P}^*]{\textrm{H}}$ is polynomial, and $\sqrt[\mathscr{P}^*]{\textrm{H}}=\mathbb{F}[h_1,\dotsc,h_n]$ if and only if

$\textrm{H}=\mathbb{F}[h_1,\dotsc,h_{n_0},h_{n_0+1}^p,\dotsc,h_{n_1}^p, h_{n_1+1}^{p^2},\dotsc,h_{n_e}^{p^e}],$

where $n_e=n$ and $n_i=\dim_{\mathbb{F}}(W_0\oplus\dotsb\oplus W_i)$.