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• # Artykuł - szczegóły

## Colloquium Mathematicum

2011 | 122 | 2 | 275-287

## Théorème de la clôture lq-modulaire et applications

FR EN

### Abstrakty

EN
Let K be a purely inseparable extension of a field k of characteristic p ≠ 0. Suppose that $[k:k^{p}]$ 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.

275-287

wydano
2011

### Twórcy

autor
• Département de mathématiques, Faculté des sciences, Université Mohammed 1, Oujda, Maroc
autor
• Département de mathématiques, Faculté des sciences, Université Mohammed 1, Oujda, Maroc