On the Galois embedding problem for \(p\)-extensions in characteristic \(p\) (Q1209833)

The general Galois embedding problem asks whether or not a given finite Galois extension of fields \(K/k\) is embeddable in a tower of Galois extensions \(k\subset K\subset L\) in such a manner that the corresponding extension of finite groups matches a pre-designated group extension. In this paper a special case of the problem is examined. Here the field characteristic is \(p>0\) and the group extension has kernel \(\mathbb{Z}/p\mathbb{Z}\); otherwise the fields are unrestricted. The author gives a condition in order that a Galois extension followed by an Artin-Schreier extension be Galoisian. The main result of the paper gives a criterion for the Galois embeddability, namely that either the given group extension is not split, or the base field \(k\) contains elements not expressible as \(u^ p-u\) with \(u\in K\).