Procedures for obtaining primitive polynomials and m-sequences over GF(q/sup m/) in terms of primitive polynomials and m-sequences over GF(q) are presented. Using a degree mn primitive polynomial g(x) in GF(g, x), an m-sequence over GF(q/sup m/) can be expressed as a vector m-sequence whose component m-sequences are shifted versions of the m-sequence generated by g(x). The degree-n primitive polynomials in GF(q/sup m/,x) with root alpha q/sup i/, that are factors of g(x) with root alpha when g(x) is viewed in GF(q/sup m/,x), are then developed from the m-sequence over GF(q/sup m/). Expressions for the shifts and corresponding primitive polynomial for the m-sequence generated by the uth decimation of the m-sequence generated by the polynomials are also given. Expressions for the shifts and corresponding primitive polynomial factors of g(x) for different bases expressing GF(q/sup m/) over GF(q) are presented.>