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

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