The centralizer of a semi-linear transformation

Let V be a finite-dimensional vector space over a division ring D, where D is finite-dimensional over its center F. Suppose T is a semi-linear transformation on V with associated automorphism σ of D. The centralizer of T is the ring C(T) of all linear transformations on V which commute with T. If σ^r is the identity on D for some r ? 1 and no smaller positive power of σ is an inner automorphism, then the center of C(T) is computed to be polynomials in T^r with coefficients from F₀, where F₀ is the subfield of F left elementwise fixed by σ. A matrix version of this theorem is also given.