Théorème de la clôture $lq$-modulaire et applications

Mustapha Chellali¹ ; El hassane Fliouet¹

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

Colloquium Mathematicum, Tome 122 (2011) no. 2, pp. 275-287

Résumé

Let $K$ be a purely inseparable extension of a field $k$ of characteristic $p\not=0$. 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 $K_1$ and $K_2$ of $K/k$ such that $K_1/k$ is $lq$-modular over $k$, $K_1(K_2)/K_2$ 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. $m\not = K$). More precisely if $K/k$ is infinite, then $K/m$ is infinite.

Détail

DOI : 10.4064/cm122-2-13

Mots-clés : purely inseparable extension field characteristic suppose finite recall lq modular modular finite extension moreover there exists smallest extension resp resp lq modular main result states existence greatest lq modular relatively perfect subextension other results summarized following product lq modular extensions lq modular nbsp augment ground field lq modular extension lq modularity preserved generally intermediate fields lq modular lq modular successive application theorem lq modular closure main result deduce smallest extension lq modular non trivial precisely infinite infinite