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

TL;DR

This paper investigates the irrationality of roots for seven key families of polynomials, establishing the existence of real roots and analyzing their algebraic properties using elementary number theory.

Contribution

It provides new results on the irrationality of roots for these polynomial families, combining algebraic and number-theoretic methods.

Findings

Most polynomial families have real roots.

Certain roots are proven to be irrational.

Methods involve elementary arithmetical properties of algebraic numbers.

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.