DEDICATED TO CHAO KO FOR HIS 90TH BIRTHDAYWe exhibit a deterministic algorithm for factoring polynomials in one variable over "nite "elds. It is e$cient only if a positive integer k is known for which I (p) is built up from small prime factors; here I denotes the kth cyclotomic polynomial, and p is the characteristic of the "eld. In the case k"1, when I (p)"p!1, such an algorithm was known, and its analysis required the generalized Riemann hypothesis. Our algorithm depends on a similar, but weaker, assumption; speci"cally, the algorithm requires the availability of an irreducible polynomial of degree r over Z/pZ for each prime number r for which I (p) has a prime factor l with l,1 mod r. An auxiliary procedure is devoted to the construction of roots of unity by means of Gauss sums. We do not claim that our algorithm has any practical value.2000 Academic Press