s-quasi-modular closure of a finite purely inseparable extension

Abstract

Given a field k of characteristic p\neq 0. Let K /k be a finite purely inseparable field extension of j-th exponent e_j. Recall that K is modular over k if and only if for any n \in \mathbb{N}, K^{p^n} and k are linearly disjoint over K^{p^n} \cap k. This notion, which plays a central role in the development of the Galois theory relating to purely inseparable extensions, was used by M. E. Sweedler to characterize purely inseparable extensions of bounded exponent which were tensor products of simple extensions. Since then, many authors have studied various properties of modular field extensions, including the existence of modular closures. Similarly, K /k is said to be s-quasi-modular if for all i \in \{1,…, e_s\}, K^{p^i} and k are K^{p^i} \cap k linearly disjoint. Motivated by R. Rasala’s work, We characterize the notion of s-quasi modularity and we then a method which makes it possible to build the s-quasi modular closure of K /k. In particular, if s = n, we find the Rasala result.

Keywords:  Purely inseparable extension, modular extension, s-quasi-modular.

MSC numbers: Primary 12F15.

Downloads: Full-text PDF