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

Authors : El Hassane FLIOUET

Pages : 100-120

Doi:10.24330/ieja.1058420

View : 106 | Download : 16

Publication Date : 2022-01-17

Article Type : Research Paper

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 insert ignore into journalissuearticles values(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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