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, K _F ), where K _F denotes the kernel of $(F,+, \cdot)$. In case that K is a commutative subfield of K _F such that dim(F, K) is finite, it makes sense to investigate the characteristic polynomial of λ _a 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.