# en fr $infty$-$w 0$-generated field extensions Extensions ∞-w 0 -générées

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Abstract : In this note, we continue to be interested in the relationship that connects the restricted distribution of finitude at the local level of intermediate fields of a purely inseparable extension $K-k$ to the absolute or global finitude of $K-k$. In -{\it $w 0$-generated field extensions,}Arch. Math. {\bf 47}, 1986, 410-412-, JK Deveney constructed an example of modular extension $K-k$ called $w 0$-generated such that for any proper subfield $L$ of $K-k$, $L$ is finite over $k$, and for every $n \in {\mathbf N}$, we have $k^{p^{- n}} \cap K: k = p^{2n}$. This example has proved to be extremely useful in the construction of other examples of $w 0$-generated extensions. In particular, we prolong the $w 0$-generated to an extension of unspecified finite size.However, when $K-k$ is of unbounded size, we show that any modular extension of unbounded exponent admits a proper subextension of unbounded exponent. This brings us to study the $w 0$-generated in the restricted sense. In addition, with the aim of extending the $w 0$-generated to a purely inseparable extension of unbounded size, we propose other generalizations.

Keywords : $w 0$-generated field extensions Modular extension Degree of irrationality Purely inseparable $+\infty$-$w 0$-generated $q$-finite extension

Autor: El Hassane Fliouet - Fliouet Résumé -

Fuente: https://hal.archives-ouvertes.fr/

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