This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GF ( p m ) {\text {GF}}({p^m}) to the problem of finding the roots in GF ( p ) {\text {GF}}(p) of certain other polynomials over GF ( p ) {\text {GF}}(p) . The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the logarithm of the order of the finite field. Certain observations on the application of these methods to the factorization of polynomials over the rational integers are also included.