Purely nonseparable extensions with unbounded exponent. (Extensions purement inséparables d’exposant non borné.) (French) Zbl 1122.12002

Let \(K\) be a purely inseparable extension of a field \(k\) of characteristic \(p\neq 0\). M. E. Sweedler showed that if \(K\) over \(k\) is of finite exponent (i.e., there is a positive integer \(n\) such that \(K^{p^n}\subset k\)) then \(K\) is modular over \(k\) (i.e., \(K^{p^i}\) and \(k\) are linearly disjoint for all positive integers \(i\)) if and only if \(K\) is isomorphic to the tensor product of simple extensions of \(k\).
An attempt to find an analogue to this theorem in the infinite exponent case has been made by L. A. Kime, who extended the definition of a simple extension to include those of the form \(k[x,x^{1/p},x^{1/p^2},\cdots ]\). She showed that if \(K\) is isomorphic to the tensor product of these simple extensions of \(k\), then \(K\) is modular over \(k\), but that the converse is false even with the restriction \([k\colon k^p]<\infty \).
In this paper, the authors introduce and studiy two other generalizations of a simple extension (called 2-simple and 3-simple). The final part of the paper contains an interesting example of a class of modular extensions \(K/k\) such that \(K\) is not a tensor product of 3-simple extensions over a finite extension \(L\) of \(k\).


12F15 Inseparable field extensions
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