Counting irreducible polynomials over finite fields

Irreducible Polynomials over Finite Fields

As we will see, modular arithmetic aids in testing the irreducibility of polynomials and even in completely factoring polynomials in Z[x]. If we expect a polynomial f(x) is irreducible, for example, it is not unreasonable to try to find a prime p such that f(x) is irreducible modulo p. If we can find such a prime p and p does not divide the leading coefficient of f(x), then f(x) is irreducible ...

Construction of Irreducible Polynomials over Finite Fields

In this paper we investigate some results on the construction of irreducible polynomials over finite fields. Basic results on finite fields are introduced and proved. Several theorems proving irreducibility of certain polynomials over finite fields are presented and proved. Two theorems on the construction of special sequences of irreducible polynomials over finite fields are investigated in de...

Constructing irreducible polynomials over finite fields

We describe a new method for constructing irreducible polynomials modulo a prime number p. The method mainly relies on Chebotarev’s density theorem.

Generators and irreducible polynomials over finite fields

Weil’s character sum estimate is used to study the problem of constructing generators for the multiplicative group of a finite field. An application to the distribution of irreducible polynomials is given, which confirms an asymptotic version of a conjecture of Hansen-Mullen.

Counting Reducible, Powerful, and Relatively Irreducible Multivariate Polynomials over Finite Fields