I am trying to understand the proof of Gauss's Lemma as given in Dummit and Foote Section 9.3 pages 303-304 (see attached)
On page 304, part way through the proof, D&F write:
"Assume d is not a unit (in R) and write d as a product of irreducibles in R, say . Since is irreducible in R, the ideal is prime (cf Proposition 12, Section 8.3 - see attached) so by Proposition 2 above (see attached) the ideal is prime in R[x] and is an integral domain. ..."
My problems with the D&F statement above are as follows:
(1) I cannot see why the ideal is a prime ideal. Certainly Proposition 12 states that "In a UFD a non-zero element is prime if and only if it is irreducible" so this means is prime since we were given that it was irreducible. But does that make the principal ideal a prime ideal? I am not sure! Can anyone show rigorously that a prime ideal?
(2) Despite reading Proposition 12 in Section 8.3 I cannot see why the ideal is prime in R[x] and is an integral domain. ...". (Indeed, I am unsure that is an ideal!) Can anyone show explicitly and rigorously why this is true?
I would really appreciate clarification of the above matters.