### Abstract

We discuss a transform on the set of rational functions over the finite field F_{q}. For a subclass of these functions, the transform yields a polynomial and its factorization as a product of the set of monic irreducible polynomials all of which share a common property P that depends on the choice of rational function. A general formula is derived from the factorization for the number of monic irreducible polynomials of degree n having property P. However it is also possible in some instances to exploit the properties of the factorization to obtain a "closed" form of the answer more directly. We illustrate the method with four examples, two of which appear in the literature. In particular, we give alternative proofs for a result of L. Carlitz on the number of monic irreducible self-reciprocal polynomials and a remarkable result of S. D. Cohen on the number of (r,m)-polynomials, that is, monic irreducible polynomials of the form f(x^{r}) of degree mr. We also give a generalization of the factorization of x^{q-1} -1 over F_{q} that includes the factorization of x^{(q-1)2} -1. The new results concern translation invariant polynomials, which lead to a consideration of the orders of elements in F_{q}, the algebraic closure of F _{q}. We show that there are an infinite number of θ ε F̄_{q} such that ord(θ) and ord(r(θ)) are related, in the sense that given one, one can infer information about the other.

Original language | English (US) |
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Pages (from-to) | 121-143 |

Number of pages | 23 |

Journal | Journal of Combinatorial Mathematics and Combinatorial Computing |

Volume | 72 |

State | Published - Feb 1 2010 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Journal of Combinatorial Mathematics and Combinatorial Computing*,

*72*, 121-143.