On explicit factors of cyclotomic polynomials over finite fields

Authors:
Liping Wang;Qiang Wang
Affiliations:
Center for Advanced Study, Tsinghua University, Beijing, China 100084;School of Mathematics and Statistics, Carleton University, Ottawa, Canada K1S 5B6
Venue:
Designs, Codes and Cryptography
Year:
2012

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Abstract

We study the explicit factorization of 2 n r-th cyclotomic polynomials over finite field $${\mathbb{F}_q}$$ where q, r are odd with (r, q) = 1. We show that all irreducible factors of 2 n r-th cyclotomic polynomials can be obtained easily from irreducible factors of cyclotomic polynomials of small orders. In particular, we obtain the explicit factorization of 2 n 5-th cyclotomic polynomials over finite fields and construct several classes of irreducible polynomials of degree 2 n---2 with fewer than 5 terms.