### Abstract

In this paper we establish necessary and sufficient conditions for D_{n}(x, a) + b to be irreducible over F_{q}, where a, b ∈ F_{q}, the finite field F_{q} of order q, and D_{n}(x, a) is the Dickson polynomial of degree n with parameter a ∈ F_{q}. As a consequence we construct several families of irreducible polynomials of arbitrarily high degrees. In addition we show that the following conjecture of Chowla and Zassenhaus from 1968 is false for Dickson polynomials: if f(x) has integral coefficients and has degree at least two, and p is a sufficiently large prime for which f(x) does not permute F_{p}, then there is an element c ∈ F_{p} so that f(x) + c is irreducible over F_{p}. We also show that the minimal polynomials of elements which generate one type of optimal normal bases can be derived from Dickson polynomials.

Original language | English (US) |
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Pages (from-to) | 118-132 |

Number of pages | 15 |

Journal | Journal of Number Theory |

Volume | 49 |

Issue number | 1 |

DOIs | |

State | Published - Oct 1994 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory