A Class of Irreducible Polynomials

For any polynomial

f (X) ∈ Q[X], write f (X) in the form f(x) =a₀+a₁X+....+a_dX^d / q

with q,a₀,a₁,......,aa_d ∈ Z,q > 1,q as small as possible, then set

           H(f) = max{|a₀|,|a₁|,......,|a_d|,q}

we show that for any relatively prime polynomials f(x), g(x) ∈ Q[X] with deg f < deg g = d, and any prime number p>2d^d H(f)^d+1 H(g)^3d, the polynomial f(X) + pg(X) is irreducible over Q. we also condider the more general case of polynomial defined over a number field.