I am reading Dummit and Foote Section 9.5 Polynomial Rings Over Fields II and need some help and guidance with the proof of Proposition 15.
Proposition 15 reads as follow:
Proposition 15. The maximal ideals in F[x] are the ideals (f(x)) generated by irreducible polynomials. In particular, F[x]/(f(x)) is a field if and only if f(x) is irreducible.
Dummit and Foote give the proof as follows:
Proof: This follows from Proposition 7 of Section 8.2 applied to the Principal Ideal Domain F[x].
My problem - can someone show me how Proposition 15 above follows from Proposition 7 of Section 8.2 (see below for Proposition 7 of Section 8.2)
I would be grateful for any help or guidance in this matter.
Dummit and Foote - Section 8.2 - Proposition 7
Proposition 7. Every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.