Polinômios irredutíveis sobre corpos finitos: construção, contagem e estimativas assintóticas

Abstract

Let $\mathbb {F}_{q} $ be a finite field with $ q $ elements. In this thesis, we focus on two types of problems about irreducible polynomials. The first one is the construction of irreducible polynomials from the composition of an irreducible polynomial with the polynomial $x^n $. This is a particular case of a more general problem about finding irreducible polynomial factorization, when it composes $f(x)$ with another polynomial to which its factorization is complete known. In particular, imposing some conditions on $ q $, $ n $, the order and the degree of the polynomial $f$, we find a procedure, which can be computationally implemented in order to determine explicitly the irreducible factors of this composition $f(x^n)$. In addition, an explicit formula for the number of irreducible factors is determined in the process. This result generalizes the results found in \cite{BGM}, \cite{Mey}, \cite{BGO} and \cite{WYF}.\\ Consequently, in the case when $f (x) = x-1$, the number of irreducible factors of $x^n-1$ is also the number of normal elements of the extension $ \mathbb {F} _ {q^n }$ over $\mathbb {F}_q$.\\ In the second part this these, we restringe our study to irreducible binomials, because there is a classic irreducibility criterion for this type of polynomial. This criterion was explored by Heyman and Shparlinski in \cite {HeSh} to find upper and lower bounds for the total number of binomials over $ \mathbb {F} _q $ with degree $ t\le T $, where $ T $ is large enough. In their work, they also find upper and lower bounds for the total number of irreducible binomials of degree $t$ over the field $ \mathbb {F} _q $ when $ q $ is limited by a constant $ Q $, but we think that this type of estimate is not very interesting because they count objects that are belong fields with different characteristics. Thus, in this second part, we determine formulas, which are asymptotically correct, for the number of irreducible binomials over $ \mathbb {F}_q $ and degree less than $ T $. This formulas substantially improves the result of Heyman and Shparlinski. Also found formulas for upper and lower bounds that are valid for small values of $T$.