On \(p\)-extensions of fields of characteristic \(p\) (Q1780411)

It is proved that a Galois extension of a field of characteristic \(p\) is completely determined by its Galois group and the endomorphism of the additive group of the group algebra that corresponds to the raising to the power \(p\). In more detail, let \(\varphi\) be an isomorphism of a field \(k\) of characteristic \(p\) onto a subfield of a field \(K\) such that \(K/\varphi(k)\) is a finite Galois extension with Galois group \(G\). Convert \(K\) to an object of the category \(\mathcal M_{k,G}\), taking for \(F\) the endomorphism \(x \to x^p\) of the additive group of the field \(K\) and setting \(x\left(\sum_{g\in G}a_gg\right) =\sum_{g\in G}\varphi(a_g)x^g\) \((x\in K, a_g\in k)\). We denote this object by \(\mathcal F^\varphi_k (K)\). If the field \(K\) is itself an extension of the field \(k\) and \(\varphi\) is the identity isomorphism, then we omit it in our notation and denote the corresponding object of the category \(\mathcal M_{k,G}\) by \(\mathcal F_k(K)\). Then the following is proved: Theorem 1. Let \(k\) be a field of characteristic \(p\) and \(G\) be a finite \(p\)-group. Let \(K\) and \(L\) be Galois extensions of the field \(k\) with Galois group \(G\). If the objects \(\mathcal F_k(K)\) and \(\mathcal F_k(L)\) of the category \(\mathcal M_{k,G}\) are isomorphic, then the fields \(K\) and \(L\) are isomorphic. Theorem 1'. Let \(\varphi\) and \(\psi\) be isomorphisms of a field \(k\) of characteristic \(p\) onto subfields of fields \(K\) and \(L\). Assume that \(K/\varphi(k)\) and \(L/\psi(k)\) are Galois extensions with Galois group \(G\). If the objects \(\mathcal F^\varphi_k (K)\) and \(\mathcal F^\psi_k (K)\) of the category \(\mathcal M_{k,G}\) are isomorphic, then there exists an isomorphism of fields \(\eta : K \to L\) such that \(\eta\varphi = \psi\). (revised October 2009)