This paper gives an elementary deterministic algorithm for completely factoring any polynomial over GF ( q ) , q = p d {\text {GF}}(q),q = {p^d} , criteria for the identification of three types of primitive polynomials, an exponential representation for GF ( q ) {\text {GF}}(q) which permits direct rational calculations in GF ( q ) {\text {GF}}(q) as opposed to modular arithmetic over GF [ p , x ] {\text {GF}}[p,x] , and a matrix representation for GF ¯ ( p ) \overline {{\text {GF}}} (p) which admits computer computations. The third type of primitive polynomial examined permits the given representation of GF ( q ) {\text {GF}}(q) to display a primitive normal basis over GF ( p ) {\text {GF}}(p) . The techniques developed require only the usual addition and multiplication of square matrices over GF ( p ) {\text {GF}}(p) . Partial tables from computer programs based on certain of these results will appear in later papers.