ON STABLE QUADRATIC POLYNOMIALS

We recall that a polynomial f (X) is an element of K[X] over a field K is called stable if all its iterates are irreducible over K. We show that almost all monic quadratic polynomials f (X) is an element of Z[X] are stable over Q. We also show that the presence of squares in so-called critical orbits of a quadratic polynomial f (X) is an element of Z[X] can be detected by a finite algorithm; this property is closely related to the stability of f (X). We also prove there are no stable quadratic polynomials over finite fields of characteristic 2 but they exist over some infinite fields of characteristic 2.