I am reading Dummit and Foote on Irreducibility in Polynomial Rings. In particular I am studying Proposition 12 in Section 9.4 Irreducibility Criteria.
Proposition 12 reads as follows: Let I be a proper ideal in the integral domain R and let p(x) be a non-constant monic polynomial in R[x]. If the image of p(x) in (R/I)[x] cannot be factored in (R/I)[x] into two polynomials of smaller degree, then p(x) is irreducible in R[x].
The proof in D&F reads as follows:
Proof: Suppose p(x) cannot be factored in (R/I)[x]. This means that there are monic non-constant polynomials a(x) and b(x) in R[x] such that p(x) = a(x) b(x). By Proposition 2, reducing the coefficients modulo I gives a factorization in (R/I)[x] with non-constant factors, a contradiction
BUT! Proposition 2 shows that there is an isomorphism between R[x]/I and and (R/I)[x]. How does this guarantee a factorization of p(x) in (R/I)[x]?
For your information Proposition 2 in D&F reads as follows: Proposition 2. Let I be an ideal of the ring R and let (I) = I[x] denote the ideal of R[x] generated by I (the set of polynomials with coefficients in I).
Then
In particular, if I is a prime ideal of R then (I) is a prime ideal of R[x].