IJPAM: Volume 97, No. 1 (2014)

$x^{2^{a}p^{b}r^{c}}-1$ OVER A FINITE FIELD

Fen Li$^1$, Xiwang Cao$^2$
$^{1,2}$College of Science
Nanjing University of Aeronautics and Astronautics
Jiangsu, 210016, P.R. CHINA

Abstract. Let $\mathbb{F}_q$ be a finite field of odd order $q$. In this paper, the irreducible factorization of $x^{2^{a}p^{b}r^{c}}-1$ over $\mathbb{F}_q$ is given in a very explicit form, where $a,b,c$ are positive integers and $p,r$ are odd prime divisors of $q-1$. It is shown that all the irreducible factors of $x^{2^{a}p^{b}r^{c}}-1$ over $\mathbb{F}_{q}$ are either binomials or trinomials. In general, denote by $v_p(m)$ the degree of prime $p$ in the standard decomposition of the positive integer $m$. Suppose that every prime factor of $m$ divides $q-1$, one has (1) if $v_p(m)\leq v_p(q-1)$ holds true for every prime number $p\vert q-1$, then every irreducible factor of $x^m-1$ in $\mathbb{F}_q$ is a binomial; (2) if $q\equiv3({\rm mod}\ 4)$, then every irreducible factor of $x^m-1$ is either a binomial or a trinomial.

Received: June 26, 2014

AMS Subject Classification: 11T06

Key Words and Phrases: irreducible factorization, binomial, trinomial

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DOI: 10.12732/ijpam.v97i1.7 How to cite this paper?

International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 97
Issue: 1
Pages: 67 - 77

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