Normal division algebras over a modular field

Abstract

Listen

Translate article

and that there exist inseparable extensions F(x) of F if and only if some quantity a of F is not the pth power of any quantity of F. An infinite field F is called perfect if either F is non-modular or every quantity of F has the form fP where p is the characteristic of F and f is in F. In any consideration of normal division algebras D over F the property that F is perfect is used only when we consider quantities of D and the minimum equations of these quantities. But if the degree n of D is not divisible by the characteristic p of F, then the assumption that F is perfect evidently has no value and is a needless extremely strong restriction on F. In most of the papers on the structure of normal division algebras written recently in Germanyl, the assumption has been that F is perfect. But I shall prove here that if F is perfect of characteristic p, then n is not divisible by p. Hence it is now necessary to consider algebras of degree pe over F of characteristic p, where F is not perfect. I shall give here a brief discussion of the validity of the major results on algebras over non-modular fields when F is assumed to be merely any infinite field. Moreover, I shall determine all normal division algebras of degree two over F of characteristic two, of degree three over F of characteristic three.? 2. The existence of a maximal separable sub-field of A. Let A be any normal division algebra of degree n over any field F, and let