In this work, a new construction based on companion matrices of primitive polynomials is provided. Given two primitive polynomials over the finite fields (Formula presented.) and (Formula presented.), we construct a ring isomorphism that transforms the companion matrix of the primitive polynomial over (Formula presented.) into a matrix with elements in (Formula presented.) whose characteristic polynomial is another primitive polynomial over (Formula presented.). © 2017 Informa UK Limited, trading as Taylor & Francis Group18