### 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) |
---|---|

Pages (from-to) | 118-132 |

Number of pages | 15 |

Journal | Journal of Number Theory |

Volume | 49 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1994 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

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*Journal of Number Theory*, vol. 49, no. 1, pp. 118-132. https://doi.org/10.1006/jnth.1994.1086

**Dickson polynomials and irreducible polynomials over finite fields.** / Gao, Shuhong; Mullen, Gary Lee.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Dickson polynomials and irreducible polynomials over finite fields

AU - Gao, Shuhong

AU - Mullen, Gary Lee

PY - 1994/1/1

Y1 - 1994/1/1

N2 - In this paper we establish necessary and sufficient conditions for Dn(x, a) + b to be irreducible over Fq, where a, b ∈ Fq, the finite field Fq of order q, and Dn(x, a) is the Dickson polynomial of degree n with parameter a ∈ Fq. 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 Fp, then there is an element c ∈ Fp so that f(x) + c is irreducible over Fp. We also show that the minimal polynomials of elements which generate one type of optimal normal bases can be derived from Dickson polynomials.

AB - In this paper we establish necessary and sufficient conditions for Dn(x, a) + b to be irreducible over Fq, where a, b ∈ Fq, the finite field Fq of order q, and Dn(x, a) is the Dickson polynomial of degree n with parameter a ∈ Fq. 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 Fp, then there is an element c ∈ Fp so that f(x) + c is irreducible over Fp. We also show that the minimal polynomials of elements which generate one type of optimal normal bases can be derived from Dickson polynomials.

UR - http://www.scopus.com/inward/record.url?scp=0011289576&partnerID=8YFLogxK

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U2 - 10.1006/jnth.1994.1086

DO - 10.1006/jnth.1994.1086

M3 - Article

AN - SCOPUS:0011289576

VL - 49

SP - 118

EP - 132

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 1

ER -