On $+\infty$-$\omega_0$-generated field extensions
On $+\infty$-$\omega_0$-generated field extensions
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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- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI