On Matrix Equations for Galois Extensions of Fields with Characteristic p

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  • On matrix equations for Galois extensions of fields with characteristc p

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Let k be a field with characteristic p and K be a Galois extension of k, the order of the Galois group being a power of p. When K is cyclic of order p over k, it is well known that equations of Artin-Schreier type are essential for its theory and this result was extended to a more general cyclic case by Albert and Witt [1], [5]. The construction of Galois extensions over k for non-abelian case was first studied by Witt [6], but he did not give equations which characterize these extensions. So it is desirable to consider this problem anew. In the case when the characteristic of k does not divide the order of the Galois group the theory on Galois algebras initiated by Hasse [4] contributed much to the study of Galois extensions, but it does not seem to be applicable for the present case. So we consider this problem apart from his theory. Our method is based on representations of Galois groups and we show that matrix equations of a certain type characterize those extensions mentioned above.

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