# $+\infty$-$w\_0$-generated field extensions

## TL;DR

This paper investigates the properties of $w_0$-generated purely inseparable field extensions, exploring their limitations and generalizations, especially in the context of unbounded extension sizes and exponents.

## Contribution

It extends the concept of $w_0$-generated extensions to unbounded size cases and proposes new generalizations for purely inseparable extensions.

## Key findings

- Unbounded size modular extensions have proper subextensions of unbounded size.
- $w_0$-generated extensions can be prolonged to larger extensions.
- New generalizations of $w_0$-generated extensions are proposed.

## 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.