Irreducibility criteria for polynomials

In this thesis, after giving a general overview of the existing criteria in the literature on the irreducibility of polynomials with integer coefficients over ℚ and ℤ, we study the irreducibility of the composition of polynomials and irreducibility of quadratic and cubic polynomials as special cases. We determine the necessary and sufficient conditions that the coefficients of a quadratic polynomial should satisfy for the polynomial to be irreducible. Then we expressed these conditions in other forms in detail. Moreover, we obtained various new results about the irreducibility of the iterates of a quadratic polynomial by making analysis on the conditions that the coefficients of the polynomial should satisfy. A detailed study of the irreducibility of cubic polynomials can be done like how it is done for quadratic polynomials in this thesis. We think that this can make important contributions to the problem of determining if a given smooth cubic curve has a rational point, i.e. if a given smooth cubic curve is an elliptic curve.