Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields

Abstract

Abstract. We give, over a finite field F q , explicit factorizations into a product of irreducible polynomials, of the cyclotomic polynomials of order 3 · 2 n , the Dickson polynomials of the first kind of order 3 · 2 n and the Dickson polynomials of the second kind of order 3 · 2 n − 1.

On explicit factors of cyclotomic polynomials over finite fields

“…In this paper we obtain the complete factorization of Φ 2 n r over F q for arbitrary r ≥ 3 odd and q odd such that gcd(q, r) = 1. Thus, we generalize the results in [11] and [26]. We make the assumption that the explicit factorization of Φ r is given to us as a known.…”

“…The factorization of Φ 2 n over F q when q ≡ 1 (mod 4) can be found for example in [15] and is stated here in Theorem 3.10; the more difficult case when q ≡ 3 (mod 4) was achieved in 1996 by Meyn [16]. More recently, Fitzgerald and Yucas (2007) [11] gave the factorization of Φ 2 n r over F q for the special cases where r is an odd prime and q ≡ ±1 (mod r) is odd. As a result, the factorizations over F q of Φ 2 n 3 , and the Dickson polynomials of the first and second kind D 2 n 3 , E 2 n 3−1 , respectively, are thus obtained.…”

Explicit idempotents of finite group algebras

“…In general, the factorization of Φ d (x) in F q [x] for arbitrary d and q is an open problem. Some especial cases can be found in [5], [6] and [7].…”

Explicit Factorization of Cyclotomic Polynomials Over Finite Fields

“…The case for q ≡ 3(mod 4) is done in [1] by Helmut Meyn. Fitzgerald and Yucas [5] have studied the explicit factorization of Q 2 n .r (x), where r is a prime and q ≡ ±1(mod r) over F q in order to obtain the explicit factorization of Dickson polynomial which results to the explicit factorization of Q 2 n .3 (x) and Dickson polynomial D 2 n .3 (x) over F q . Recently, Liping Wang and Qiang Wang [3] have shown that all irreducible factors of cyclotomic polynomial can be obtained easily from irreducible factors of cyclotomic polynomial of small orders and also studied the explicit factorization of 2 n .5-th cyclotomic polynomial over finite fields which give rise to the construction of several classes of irreducible polynomial of degree 2 n−2 with fewer than 5 terms.…”

“…In this paper, we study the factors of the cyclotomic polynomial Q 2 n .7 (x) over finite field F q when q ≡ ±1, ±2, ±3(mod 7) as an extension of [3] and [5].…”