Deterministic Irreducibility Testing of Polynomials over Large Finite Fields

Fast Parallel Absolute Irreducibility Testing

We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the cla...

Factoring and Testing Irreducibility of Sparse Polynomials over Small Finite Fields

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...

Testing polynomial irreducibility without GCDs

We determine classes of degrees where testing irreducibility for univariate polynomials over finite fields can be done without any GCD computation. This work was partly funded by the INRIA Associate Team “Algorithms, Numbers, Computers” (http://www.loria.fr/~zimmerma/anc.html). Key-words: finite fields, irreducible polynomial, GCD ∗ ANU, Canberra, [email protected] Test d’irréductibilité de polynôme...

On the Reducibility of Cyclotomic Polynomials over Finite Fields

The irreducibility of a polynomial in a prime modulus implies irreducibility over the rationals. However, the converse is certainly not true. In fact, there are some polynomials that are irreducible over the rationals yet reducible in every prime modulus. We show that the nth cyclotomic polynomial reduces modulo all primes if and only if the discriminant of the nth cyclotomic polynomial is a sq...