Factoring in ℚ[x], I

Abstract

In Chapters 14 and 15 we looked at polynomials with real or complex coefficients. We completely determined the irreducible polynomials, and showed how to make a start at locating the roots of a real polynomial. In this chapter we begin considering the same questions for polynomials with coefficients in ℚ, the field of rational numbers. Here the situation is much different from the situation over ℝ or ℂ. Over ℚ there are many irreducible polynomials of any degree, and determining which polynomials are irreducible is difficult, compared to the real or complex case. On the other hand, finding roots (and therefore irreducible factors of degree 1) of a polynomial in ℚ [x] is easy, and we will eventually give two different explicit procedures for determining the complete factorization of any polynomial with rational coefficients in a finite number of steps.