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:

This follows from Proposition 7 of Section 8.2 applied to the Principal Ideal Domain F[x].Proof:

- 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)

My problem

I would be grateful for any help or guidance in this matter.

Peter

Dummit and Foote - Section 8.2 - Proposition 7

Proposition 7. Every nonzero prime ideal in a Principal Ideal Domain is a maximal ideal.