On additive polynomials over a finite field

This paper is based on the interpretation of the ring A(q) of additive poly nomials in one variable over a finite field F-q as a maximal R-order inside a certain skew-field D, R being a principal ideal domain isomorphic to F-p [T]. The classical (1930's) structure theory of maximal orders in global fi elds is used to solve enumeration questions involving the iteration of memb ers of A(q).