On the preservation for Quasi-Modularity of field extensions

Abstract

Let k be a field of characteristic p ≠ 0. In 1968. M. H. Swecdler revealed for the first time, the usefulness of the concept of modularity. This notion, which plays an important role especially for Galois theory of purely inseparable extensions, was used to characterize purely inseparable extensions of bounded exponent which were tensor products of simple extensions. A natural extension of the definition of modularity is to say that K/k is q-modular (quasi modular) if K is modular up to some finite extension. In subsequent papers. M. Chel- lali and the author have studied various properly of q-modular field extensions, including the questions of q-modularity preservation in case [k : kᵖ] is finite. This paper grew out of an attempt to find analogue results concerning the preservation of q-modularity, without the hypothesis on k but with extra assumptions on K/k. In particular, we investigate existence conditions of lower (resp. upper) quasi-modular closures for a given q-finite extension.