Irreducibility and r-th root finding over finite fields

Irreducibility and Deterministic r-th Root Finding over Finite Fields

Constructing r -th nonresidue over a €nite €eld is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree r e (where r is a prime) over a given €nite €eld Fq of characteristic p (equivalently, constructing the bigger €eld Fqr e ). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some ne...

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...

Abelian Extensions of Arbitrary Fields

Let k be an Hilbertian field, i.e. a field for which Hilbert's irreducibility theorem holds (cf. [1, 5]). It is obvious that the degree of the algebraic closure k of k is infinite with respect to k. It is not obvious that the same is true for the maximal p-extension of k, p a prime number. Let A be a finite abelian group. The question whether there exists a Galoisian extension l/k with Galois g...

Factoring and Testing Irreducibility of Sparse Polynomials over Small Finite Fields

Deterministic Irreducibility Testing of Polynomials over Large Finite Fields

We present a sequential deterministic polynomial-time algorithm for testing dense multivariate polynomials over a large finite field for irreducibility. All previously known algorithms were of a probabilistic nature. Our deterministic solution is based on our algorithm for absolute irreducibility testing combined with Berlekamp’s algorithm.