I am reading Dummit and Foote Section 9.5 Polynomial Rings Over Fields II.

I am trying to understand the proof of Proposition 16 which reads as follows:

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Proposition 16. Let F be a field. Let g(x) be a nonconstant monic polynomial of F[x] and let

be its factorization into irreducibles, where all the are distinct.

Then we have the following isomorphism of rings:

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The proof reads as follows:

Proof: This follows from the Chinese Remainder Theorem (Theorem 7.17), since the ideals and are comaximal if and are distinct (they are relatively prime in the Euclidean Domain F[x], hence the ideal generated by them is F[x]).

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My question: I can follow the reference to the Chinese Remainder Theorem but I cannot follow the argument that establishes the necessary condition that the and are comaximal.

That is, why are and comaximal - how exactly does it follow from and being relatively prime in the Euclidean Domain F[x]?

Any help would be very much appreciated.

Peter