On $+\infty$-$\omega_0$-generated field extensions

Year 2022, Volume: 31 Issue: 31, 100 - 120, 17.01.2022

El Hassane Flıouet

https://doi.org/10.24330/ieja.1058420

Abstract

A purely inseparable field extension $K$ of a field $k$ of characteristic $p\not=0$ is said to be $\omega_0$-generated
over $k$ if $K/k$ is not finitely generated, but $L/k$ is finitely generated for each proper intermediate field $L$.
In 1986, Deveney solved the question posed by R. Gilmer and W. Heinzer, which consists in knowing if the lattice of intermediate fields of an $\omega_0$-generated field extension $K/k$ is necessarily linearly ordered under inclusion, by constructing an example of an $\omega_0$-generated field extension where $[k^{p^{-n}}\cap K: k]= p^{2n}$ for all positive integer $n$. This example has proved to be extremely useful in the construction of other examples of $\omega_0$-generated field extensions (of any finite irrationality degree).
In this paper, we characterize the extensions of finite irrationality degree which are $\omega_0$-generated.
In particular, in the case of unbounded irrationality degree, any modular extension of unbounded exponent contains a proper subfield of unbounded exponent over the ground field.
Finally, we give a generalization, illustrated by an example, of the $\omega_0$-generated to include modular purely inseparable extensions of unbounded irrationality degree.

Keywords

Purely inseparable, $q$-finite extension, Modular extension, $\omega_0$-generated field extension

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