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The analysis and synthesis of polynomials and sequences over has received considerable attention in recent years with the increasing use of PN sequences. In this paper a new approach to the problem is presented in which the polynomial coefficients and the sequence digits are derived in terms of the values assumed by a special class of polynomials, called "cyclonomials," at an arbitrary primitive element of . For each value of the cyclonomials are determined by the partition of the set into cyclotomic cosets. A method of deriving all primitive polynomials of degree from a given one of the same degree is described. A short outline of an approach to the more difficult task of synthesizing an initial primitive polynomial is also presented.