Sur l'irrationalit\'e des racines de certaines familles de polyn\^omes

Abstract

We are interested in irrationality of roots for seven important families of polynomials : Tchebichef polynomials, Legendre polynomials, Laguerre polynomials, Hermite polynomials, Bessel polynomials, Bernoulli polynomials and Euler polynomials. One first proves, for the most part of them, existence of real roots then one studies the irrationality of them. The methods used are based on elementary arithmetical properties of algebraic numbers, some of them becoming from more general proofs that have allowed to derive the irreducibility of some of these polynomials.