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

V. V. Bavula

paper

Explicit descriptions of the subfields $(N L)^{p i}$ and $(N L)^{p i} (N L)^{se p}$ of $N L$ and new explicit criteria for $N L = (N L)^{p i} (N L)^{se p}$

arXiv:2606.19962

Abstract

Let $L = K (θ) ≃ K [x] / f (x)$ be a simple field extension in prime characteristic $p > 0$ , $L^{se p}$ and $L^{p i}$ be the maximal separable and purely inseparable subfields of $L$ , respectively. Let $N / K$ be a purely inseparable field extension. For the field extensions $L / K$ and $N L / N$ , 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 $m_{f, N}$ : $L^{p i}$ , $L^{p i} L^{se p}$ , $(N L)^{p i}$ and $(N L)^{p i} (N L)^{se p}$ . From these results, we derive new explicit criteria for $L = L^{p i} L^{se p}$ and $N L = (N L)^{p i} (N L)^{se p}$ .

20 pages

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

Abstract

Explicit descriptions of the subfields $(N L)^{p i}$ and $(N L)^{p i} (N L)^{se p}$ of $N L$ and new explicit criteria for $N L = (N L)^{p i} (N L)^{se p}$