Modified Cyclotomic Polynomial and Its Irreducibility

International Mathematical Forum, 2011

2011

A cyclotomic polynomial Φ n (x) is said to be ternary if n = pqr with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ pqr (x) with p < q < r, p and q fixed and r a free prime.

Acta Arithmetica, 1999

Designs, Codes and Cryptography, 2012

Let q = p s be a power of a prime number p and let Fq be the finite field with q elements. In this paper we obtain the explicit factorization of the cyclotomic polynomial Φ 2 n r over Fq where both r ≥ 3 and q are odd, gcd(q, r) = 1, and n ∈ N. Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of Φr over Fq is given to us as a known. Let n = p e 1 1 p e 2 2 • • • p es s be the factorization of n ∈ N into powers of distinct primes p i , 1 ≤ i ≤ s. In the case that the orders of q modulo all these prime powers p e i i are pairwise coprime we show how to obtain the explicit factors of Φn from the factors of each Φ p e i i. We also demonstrate how to obtain the factorization of Φmn from the factorization of Φn when q is a primitive root modulo m and gcd(m, n) = gcd(φ(m), ordn(q)) = 1. Here φ is the Euler's totient function, and ordn(q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over Fq and generalize a result due to Varshamov (1984) [23].

2019

The notion of block divisibility naturally leads one to introduce unitary cyclotomic polynomials Φ n pxq. They can be written as certain products of cyclotomic poynomials. We study the case where n has two or three distinct prime factors using numerical semigroups, respectively Bachman’s inclusion-exclusion polynomials. Given m ě 1 we show that every integer occurs as a coefficient of Φ mn pxq for some n ě 1 following Ji, Li and Moree [9]. Here n will typically have many different prime factors. We also consider similar questions for the polynomials pxn ́ 1q{Φ n pxq, the inverse unitary cyclotomic polynomials.

Involve, a Journal of Mathematics, 2011

A cyclotomic polynomial Φ n (x) is said to be ternary if n = pqr with p, q and r distinct odd primes. Ternary cyclotomic polynomials are the simplest ones for which the behaviour of the coefficients is not completely understood. Here we establish some results and formulate some conjectures regarding the coefficients appearing in the polynomial family Φ pqr (x) with p < q < r, p and q fixed and r a free prime.

2018

Let q = p e be a power of prime number p and F q be a finite field with q elements. Let Φ n be the nth cyclotomic polynomial over F q such that q is congruent to ±1 modulo each prime divisor of n. We use composed products to obtain an explicit factorization of Φ n over the finite field F q. i Dedication I would like to dedicated my thesis to my lovely family: My grateful Mother and Father for their support, my wonderful husband for his care, and my lovely sisters and brothers for their love.

Glasgow Mathematical Journal, 1985

We define the nth cyclotomic polynomial Φn(z) by the equationand we writewhere ϕ is Euler&#39;s function.Erdös and Vaughan [3] have shown thatuniformly in n as m-→∞, whereand that for every large m

Mathematics of Computation, 2011

We present three algorithms to calculate Φ n ( z ) \Phi _n(z) , the n t h n_{th} cyclotomic polynomial. The first algorithm calculates Φ n ( z ) \Phi _n(z) by a series of polynomial divisions, which we perform using the fast Fourier transform. The second algorithm calculates Φ n ( z ) \Phi _n(z) as a quotient of products of sparse power series. These two algorithms, described in detail in the paper, were used to calculate cyclotomic polynomials of large height and length. In particular, we have found the least n n for which the height of Φ n ( z ) \Phi _n(z) is greater than n n , n 2 n^2 , n 3 n^3 , and n 4 n^4 , respectively. The third algorithm, the big prime algorithm, generates the terms of Φ n ( z ) \Phi _n(z) sequentially, in a manner which reduces the memory cost. We use the big prime algorithm to find the minimal known height of cyclotomic polynomials of order five. We include these results as well as other examples of cyclotomic polynomials of unusually large height, and bo...

2008

Let Φ n (x) denote the nth cyclotomic polynomial. In 1968 Sister Marion Beiter conjectured that a n (k), the coefficient of x k in Φ n (x), satisfies |a n (k)| ≤ (p + 1)/2 in case n = pqr with p < q < r primes (in this case Φ n (x) is said to be ternary). Since then several results towards establishing her conjecture have been proved (for example |a n (k)| ≤ 3p/4). Here we show that, nevertheless, Beiter's conjecture is false for every p ≥ 11. We also prove that given any ǫ > 0 there exist infinitely many triples (p j , q j , r j ) with p 1 < p 2 < . . . consecutive primes such that |a p j q j r j (n j )| > (2/3ǫ)p j for j ≥ 1.