Exponents of skew polynomials

We introduce the notion of a relative exponent for two elements in a finite ring and apply this to define and study the exponent of a polynomial in an Ore extension of the form Fq[t;θ]Fq[t;θ]. This generalizes the classical notion of exponent (a.k.a. order or period) of a polynomial with coefficients in a finite field. The classical connections between the exponent of a polynomial, the order of its roots and of its companion matrix are obtained via the study of a notion of skew order of an element in a finite group.