The number of irreducible polynomials of degree n over Fq with given trace and constant terms

AbstractWe study the number Nγ(n,c,q) of irreducible polynomials of degree n over Fq where the trace γ and the constant term c are given. Under certain conditions on n and q, we obtain bounds on the maximum of Nγ(n,c,q) varying c and γ. We show with concrete examples how our results improve the previously known bounds. In addition, we improve upper and lower bounds of any Nγ(n,c,q) when n=a(q−1) for a nonzero constant term c and a nonzero trace γ. As a byproduct, we give a simple and explicit formula for the number N(n,c,q) of irreducible polynomials over Fq of degree n=q−1 with a prescribed primitive constant term c

On the number of NN-free elements with prescribed trace

In this paper we derive a formula for the number of $N$-free elements over a finite field $\mathbb{F}_q$ with prescribed trace, in particular trace zero, in terms of Gaussian periods. As a consequence, we derive a simple explicit formula for the number of primitive elements, in quartic extensions of Mersenne prime fields, having absolute trace zero. We also give a simple formula in the case when $Q = (q^m-1)/(q-1)$ is prime. More generally, for a positive integer $N$ whose prime factors divide $Q$ and satisfy the so called semi-primitive condition, we give an explicit formula for the number of $N$-free elements with arbitrary trace. In addition we show that if all the prime factors of $q-1$ divide $m$, then the number of primitive elements in $\mathbb{F}_{q^m}$, with prescribed non-zero trace, is uniformly distributed. Finally we explore the related number, $P_{q, m, N}(c)$, of elements in $\mathbb{F}_{q^m}$ with multiplicative order $N$ and having trace $c \in \mathbb{F}_q$. Let $N \mid q^m-1$ such that $L_Q \mid N$, where $L_Q$ is the largest factor of $q^m-1$ with the same radical as that of $Q$. We show there exists an element in $\mathbb{F}_{q^m}^*$ of (large) order $N$ with trace $0$ if and only if $m \neq 2$ and $(q,m) \neq (4,3)$. Moreover we derive an explicit formula for the number of elements in $\mathbb{F}_{p^4}$ with the corresponding large order $L_Q = 2(p+1)(p^2+1)$ and having absolute trace zero, where $p$ is a Mersenne prime

The number of irreducible polynomials of degree n over Fq with given trace and constant terms

We study the number Nγ (n, c, q) of irreducible polynomials of degree n over Fq where the trace γ and the constant term c are given. Under certain conditions on n and q, we obtain bounds on the maximum of Nγ (n, c, q) varying c and γ. We show with concrete examples how our results improve the previously known bounds. In addition, we improve upper and lower bounds of any Nγ (n, c, q) when n = a (q - 1) for a nonzero constant term c and a nonzero trace γ. As a byproduct, we give a simple and explicit formula for the number N (n, c, q) of irreducible polynomials over Fq of degree n = q - 1 with a prescribed primitive constant term c

On some classes of irreducible polynomials over finite fields

In this thesis we describe the work in literature on various aspects of the theory of polynomials over nite elds. We focus on properties like irreducibility and divisibility. We also consider existence and enumeration problems for irreducible polynomials of special types. After the introductory Chapter 1, we collect the well-known results on irreducibility of binomials and trinomials in Chapter 2, where we also present the number of irreducible factors of a xed degree k of xt due to L. Redei. Chapter 3 is on self-reciprocal polynomials. An in nite family of irreducible, self-reciprocal polynomials over F2 is presented in Section 3.2. This family was obtained by J. L. Yucas and G. L. Mullen. Divisibility of self-reciprocal polynomials over F2 and F3 is studied in Sections 3.3 and 3.4 following the work of R. Kim and W. Koepf. The last chapter aims to give a survey of recent results concerning existence and enumeration of irreducible polynomials with prescribed coefficients