On Characteristic Polynomials and the Norm Mapping for Dickson Near-Fields

Abstract

For any (left) near-field \((F,+, \cdot)\) and a ∈ F the left multiplication \(\lambda_a : x \mapsto a \cdot x (x \in F)\) is an endomorphism of the right vector space (F, KF), where KF denotes the kernel of \((F,+, \cdot)\). In case that K is a commutative subfield of KF such that dim(F, K) is finite, it makes sense to investigate the characteristic polynomial of λa and, in particular, the norm of a over K. In this note we study these objects for the class of Dickson near-fields which are coupled to commutative fields. We show how characteristic polynomials and norms of such a near-field can be derived from those of the underlying field.