The Concept of -Cycle and Applications

The concept of -cycle is investigated for its properties and applications. Connections with irreducible polynomials over a finite field are established with emphases on the notions of order and degree. The results are applied to deduce new results about primitive and self-reciprocal polynomials. 1. Introduction Let denote the finite field of elements and let , . Let be ？？distinct numbers chosen from . If then we say that forms a -cycle？mod？ with leading element , abbreviated by -cycle or -cycle when the leading element in the cycle is immaterial, and call the length of this -cycle . The notion of -cycles was introduced by Wan in his book [1, page 203]. Since , where is the order of in (the multiplicative group of nonzero integers modulo ), it clearly follows that each -cycle always has a unique length which is the least positive integer for which . Observe that -cycles are nothing but -orbits ; that is, acts on by multiplication with . The concept of -cycles is important because of the following connections with irreducible polynomials. (A) (see [1, Theorem 9.11]) Let be a primitive th root of unity (if the order of in is , then there exists a primitive th root of unity in ). If is a -cycle , then is a monic irreducible factor of in . Conversely, if is a monic irreducible factor of in , then all the roots of are powers of whose exponents form a -cycle . We henceforth refer to these two facts as the -correspondence. (B) (see [1, Corollary 9.12]) The number of distinct irreducible polynomials dividing in is equal to the number of -cycles formed with the leading elements taken from . Our objectives here are to illustrate the versatility of -cycles by using them to prove new results about irreducible polynomials. General properties of -cycles are given in the next section; specific details in two special cases corresponding to are worked out for applications to primitive and self-reciprocal polynomials in the last section. Section 3 deals with results about the order of a polynomial, while Section 4 does the same for the degree of a polynomial. Section 5 shows that knowing a -cycle is equivalent to knowing all coefficients of the corresponding polynomial (2). The last section provides applications of -cycles to primitive and self-reciprocal polynomials. Notation and Terminology. Throughout, we fix the following symbols and their meanings.(i) is a fixed prime, is a power of , and is the finite field with elements.(ii) is a fixed positive integer such that .(iii) is the number of (positive integer) divisors of .(iv) (all divisors of ).(v)For , the M？bius