On Generalized Artin-Schreier Equations

概要

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Let k be a field with characteristic p and K a finite Galois extension of k whose Galois group is denoted by G. In the previous article [1] we showed that K can be defined by matrix equations of a certain type when the order of G is a power of p and that these equations have properties similar to those of Artin-Schreier equations. The aim of the present work is first to show that the above result can be extended to the general case when G is arbitrary, and secondly to investigate relations between the form of generalized Artin-Schreier equations and the type of representations of G determined by these equations. It is hoped that our theory will contribute in some degree to answering the qestion as to how we can construct Galois extensions whose Galois groups are isomorphic to a given group.
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