A KNOT K in S’ is said to have period n > I if there is a transformation T of S’ of order n such that K is invariant under T and the fixed point set of T is a circle B, disjoint from K. (The positive solution of the Smith conjecture implies that B is unknotted and the transformation is equivalent to the one-point compactification of rotation about the z-axis in R3.) This paper discusses a variety of issues relating to the Alexander polynomial of periodic knots. Many techniques are available for determining the possible periods of a knot. The first significant results were those of Trotter [S. 273 on the periods of torus knots, obtained by analyzing possible actions on the fundamental group of the knot. Murasugi’s study [IS] of the Alexander polynomials of periodic knots proved especially powerful. Further work on the polynomial and Alexander ideals was done by Hillman [ 123. More recently. results concerning the Jones polynomial [l7, lg. 25. 261. hyperbolic structures on knot complcmcnts [I], and the geometry of 3-manifolds [7]. have been applied to the study of periodic knots as well. Further refcrcnccs include [3. 4, 6, 8. 9. IO, 16. 201. Nonetheless, conditions on the Alexander polynomial of a knot yield the most easily computable restrictions on the periods of a knot. As will be demonstrated, these methods continue to provide especially powerful tools. Our concern here is the description of both necessary and suthcient conditions for a polynomial to be the Alexander polynomial of a periodic knot. The material on suthciency is completely new. As a brief example of an application of these results, consider the knot polynomial 3f4 91’ + I lr* 91 + 3. This is the polynomial of the knot IO,,,, which does not have period 3 by [I]. We will show that in fact thsro is a period 3 knot having this Alexander polynomial. Murasugi showed that the Alcxandcr polynomial of a knot of period n satisfies certain conditions, described in detail in $I. Our focus is on two questions. one algebraic and one geometric: when does a polynomial satisfy the Murasugi conditions, and is a polynomial satisfying Murasugi’s conditions the Alcxandcr polynomial of a knot of period n? Our formulation presents the conditions in a way that yields many new corollaries. In fact, we are able to eliminate many cases of possible periods of knots, which until now have only been addressed with much more subtle invariants. (Of course, as any knot polynomial occurs as the polynomial for an infinite collection on knots, our examples represent infinite families. Other methods arc more restrictive.)