Irreducibility of the iterates of a quadratic polynomial over a field

1. Introduction. Let K be a field of characteristic p ≥ 0 and let f(X) be a polynomial of degree at least two with coefficients in K. We set f₁(X) = f(X) and define ${\displaystyle f_{r+1}\left(X\right)=f\left(f_r\left(X\right)\right)}$ for all r ≥ 1. Following R. W. K. Odoni [7], we say that f is stable over K if ${\displaystyle f_r\left(X\right)}$ is irreducible over K for every r ≥ 1. In [6] the same author proved that the polynomial f(X) = X² - X + 1 is stable over ℚ. He wrote in [7] that the proof given there is quite difficult and it would be of interest to have an elementary proof. In the sequel we shall use elementary methods for proving the stability of quadratic polynomials over number fields; especially the rational field, and over finite fields of characteristic p ≥ 3.