Explicit descriptions of the subfields $(NL{)}^{pi}$ and $(NL{)}^{pi}(NL{)}^{sep}$ of $NL$ and new explicit criteria for $NL=(NL{)}^{pi}(NL{)}^{sep}$

Let $L=K(\theta )\simeq K[x]\mathrm{/}f(x)$ be a simple field extension in prime characteristic $p>0$, $L^{sep}$ and $L^{pi}$ be the maximal separable and purely inseparable subfields of $L$, respectively. Let $N\mathrm{/}K$ be a purely inseparable field extension. For the field extensions $L\mathrm{/}K$ and $NL\mathrm{/}N$, the aim of the paper is to give explicit descriptions of the following subfields and their degrees in terms of the coefficients of the polynomial $f$ and two numerical field invariants $m_f$ and $m_{f,N}$: $L^{pi}$, $L^{pi}{L}^{sep}$, $(NL{)}^{pi}$ and $(NL{)}^{pi}(NL{)}^{sep}$. From these results, we derive new explicit criteria for $L=L^{pi}{L}^{sep}$ and $NL=(NL{)}^{pi}(NL{)}^{sep}$.