On Algebraic Groups Defined by Norm Forms of Separable Extensions

Abstract

Let K be any field, and L a separable extension of K of finite degree. L has a structure of vector space over K, and we shall denote this space by V. The space of endomorphisms of V will be denoted by Let x be any element of L, and N(x) the norm of x relative to the extension L/K. N is then a function defined on V with values in K. We shall call N the norm form on V. The multiplicative groups of non-zero elements of K and L will be denoted by K* and L* respectively. Let H be any subgroup of if K*. Then the elements z of L* such that N(z)∈H form a subgroup of L*, which we shall denote by GH. On the other hand the elements s of such that N(sx) = Λ(s)N(x) with Λ(s)∈H for all X∈V, form obviously a subgroup of GL(V), which we shall denote by becomes an algebraic group if H=K* or {1}. In case will mean the group of linear transformations of V leaving semi-invariant the norm form of L/K and in case will mean the group of linear transformations of V leaving invariant the norm form of L/K.