Irreducible Polynomials in UFDs

In general, given a field K and a polynomial \(f(X) \in K[X]\) , it is difficult to decide whether f is irreducible or not. However, this is extremely important for the further investigation of the roots of f in Galois theory. In this chapter, we develop techniques that can help us whenever the field K is the quotient field of a UFD R. For example for \(K=\mathbb {Q}\) , this is the case with \(R=\mathbb {Z}\) .