On additive polynomials over a finite field

This paper is based on the interpretation of the ring A(q) of additive polynomials 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 fields is used to solve enumeration questions involving the iteration of members of A(q).