Visibly irreducible polynomials over finite fields

H. Lenstra has pointed out that a cubic polynomial of the form (x-a)(x-b)(x-c) + r(x-d)(x-e), where {a,b,c,d,e} is some permutation of {0,1,2,3,4}, is irreducible modulo 5 because every possible linear factor divides one summand but not the other. We classify polynomials over finite fields that admit an irreducibility proof with this structure.Comment: 11 pages. To appear in the American Mathematical Monthl

Generating series for irreducible polynomials over finite fields

We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multi-degree and the case of indecomposable polynomials

Efficient indexing of necklaces and irreducible polynomials over finite fields

We study the problem of indexing irreducible polynomials over finite fields, and give the first efficient algorithm for this problem. Specifically, we show the existence of poly(n, log q)-size circuits that compute a bijection between {1, ... , |S|} and the set S of all irreducible, monic, univariate polynomials of degree n over a finite field F_q. This has applications in pseudorandomness, and answers an open question of Alon, Goldreich, H{\aa}stad and Peralta[AGHP]. Our approach uses a connection between irreducible polynomials and necklaces ( equivalence classes of strings under cyclic rotation). Along the way, we give the first efficient algorithm for indexing necklaces of a given length over a given alphabet, which may be of independent interest

On sets of irreducible polynomials closed by composition

Let $\mathcal S$ be a set of monic degree $2$ polynomials over a finite field and let $C$ be the compositional semigroup generated by $\mathcal S$. In this paper we establish a necessary and sufficient condition for $C$ to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set $\mathcal S$. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree $2$ polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions

Irreducible polynomials over finite fields

RESUMEN: Es bien sabido que cualquier cuerpo finito tiene p[elevado]n elementos con p un número primo y n un entero positivo. Recíprocamente, para cada primo p y cada entero positivo n existe un único cuerpo finito (salvo isomorfismo) con p[elevado]n elementos, Fpn. La construcción de Fpn se realiza como extensión del cuerpo primo Fp a partir de un polinomio irreducible de grado n; esto es: Fpn es isomorfo al cuerpo Fp[x]=(f(x)) con f un polinomio irreducible sobre Fp de grado n. Por la unicidad del cuerpo finito, éste también puede construirse a partir de Fpr con r un divisor de n, utilizando un polinomio irreducible sobre Fpr de grado n/r . En este trabajo de fin de grado se pretende estudiar para un cuerpo finito Fq, q = p[elevado]r, cómo construir un polinomio irreducible de grado dado que permita dar una construcción efectiva de cualquier cuerpo finito. Asimismo, se determinará el número de polinomios irreducibles existentes sobre un cuerpo finito de grado dado, así como la cantidad de polinomios irreducibles con ciertas características.ABSTRACT: It is well known that any finite field has p[elevado]n elements, where p is a prime number and n a positive integer. Reciprocally, for every prime number p and every positive integer n there exists a unique up to isomorphism finite field with pn elements: Fpn. The field Fpn is defined as an extension of the prime field Fp from an irreducible polynomial of degree n: this is, Fpn is isomorphic to the field Fp[x]=(f(x)) where f 2 Fp[x] is an irreducible polynomial of degree n. Since every finite field is unique, it can also be constructed from Fpr where r divides n using an irreducible polynomial of degree n r over Fpr . This work aims to study, for a finite field Fq with q = pn, how to build an irreducible polynomial of a xed degree that enables the efective construction of any finite field. In addition, this work will determine the number of irreducible polynomials of a xed degree over a finite field, and the number of irreducible polynomials with determined characteristics.Grado en Matemática