# Absolutely $lq$-finite extension

## TL;DR

This paper characterizes the size of purely inseparable extensions in characteristic p, providing conditions for bounded size, and explores the structure of absolutely lq-finite extensions, including a counterexample with unique properties.

## Contribution

It offers a new characterization of the size of purely inseparable extensions and constructs a novel example of an extension with specific finiteness properties.

## Key findings

- Characterization of the size of purely inseparable extensions
- Necessary and sufficient conditions for bounded size
- Construction of a counterexample with infinite size but finite intermediate fields

## Abstract

Let K/k be purely inseparable extension of characteristic p \textgreater{} 0. By invariants, we characterize the measure of the size of K/k. In particular, we give a necessary and sufficient condition that K/k is of bounded size. Furthermore, 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. Part of this problem was treated successively by J.K Devney, and in my work with M. Chellali. The other part is the subject of this paper, it is a question of describing the absolutely lq-finite extensions. Among others, any absolutely lq-finite extension decomposes into w0-generated extensions. However, we construct an example of extension of infinite size such that for any intermediate field L of K/k, L is of finite size over k. In addition, K/k does not respect the distribution of horizontal finitude.