# Absolutely lq-Finite Extensions

Absolutely lq-Finite Extensions We describe the lower quasi-finite extensions K / k $K/k$ of characteristic p > 0 $p>0$ , which are defined as follows: for every n ∈ ℕ , k p − n ∩ K / k $n\in \mathbb N, k^{p^{-n}} \cap K/k$ is finite. We are especially interested in examining the absolute case. In this regard, we give necessary and sufficient condition for an absolutely lq-finite extension to be of finite size. Moreover, we show that any extension that is at the same time modular and lq-finite is of finite size. Furthermore, we construct an example of extension K / k $K/k$ of infinite size such that for any intermediate field L of K / k , L $K/k, L$ is of finite size over k. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Vietnamica Springer Journals

# Absolutely lq-Finite Extensions

Acta Mathematica Vietnamica, Volume 44 (3) – Jun 4, 2018
29 pages

/lp/springer_journal/absolutely-lq-finite-extensions-qPHXOA2psH
Publisher
Springer Journals
Copyright © 2018 by Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd.
Subject
Mathematics; Mathematics, general
ISSN
0251-4184
eISSN
2315-4144
DOI
10.1007/s40306-018-0271-2
Publisher site
See Article on Publisher Site

### Abstract

We describe the lower quasi-finite extensions K / k $K/k$ of characteristic p > 0 $p>0$ , which are defined as follows: for every n ∈ ℕ , k p − n ∩ K / k $n\in \mathbb N, k^{p^{-n}} \cap K/k$ is finite. We are especially interested in examining the absolute case. In this regard, we give necessary and sufficient condition for an absolutely lq-finite extension to be of finite size. Moreover, we show that any extension that is at the same time modular and lq-finite is of finite size. Furthermore, we construct an example of extension K / k $K/k$ of infinite size such that for any intermediate field L of K / k , L $K/k, L$ is of finite size over k.

### Journal

Acta Mathematica VietnamicaSpringer Journals

Published: Jun 4, 2018

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