# Théorème de la clôture lq-modulaire et applications

Colloquium Mathematicae (2011)

• Volume: 122, Issue: 2, page 275-287
• ISSN: 0010-1354

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## Abstract

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Let K be a purely inseparable extension of a field k of characteristic p ≠ 0. Suppose that $\left[k:{k}^{p}\right]$ is finite. We recall that K/k is lq-modular if K is modular over a finite extension of k. Moreover, there exists a smallest extension m/k (resp. M/K) such that K/m (resp. M/k) is lq-modular. Our main result states the existence of a greatest lq-modular and relatively perfect subextension of K/k. Other results can be summarized in the following: 1. The product of lq-modular extensions over k is lq-modular over k. 2. If we augment the ground field of an lq-modular extension, the lq-modularity is preserved. Generally, for all intermediate fields K₁ and K₂ of K/k such that K₁/k is lq-modular over k, K₁(K₂)/K₂ is lq-modular. By successive application of the theorem on lq-modular closure (our main result), we deduce that the smallest extension m/k of K/k such that K/m is lq-modular is non-trivial (i.e. m ≠ K). More precisely if K/k is infinite, then K/m is infinite.

## How to cite

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Mustapha Chellali, and El hassane Fliouet. "Théorème de la clôture lq-modulaire et applications." Colloquium Mathematicae 122.2 (2011): 275-287. <http://eudml.org/doc/286235>.

@article{MustaphaChellali2011,
author = {Mustapha Chellali, El hassane Fliouet},
journal = {Colloquium Mathematicae},
keywords = {purely inseparable extension; modular extension; lq-modular extension},
language = {fre},
number = {2},
pages = {275-287},
title = {Théorème de la clôture lq-modulaire et applications},
url = {http://eudml.org/doc/286235},
volume = {122},
year = {2011},
}

TY - JOUR
AU - Mustapha Chellali
AU - El hassane Fliouet
TI - Théorème de la clôture lq-modulaire et applications
JO - Colloquium Mathematicae
PY - 2011
VL - 122
IS - 2
SP - 275
EP - 287
LA - fre
KW - purely inseparable extension; modular extension; lq-modular extension
UR - http://eudml.org/doc/286235
ER -

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