Problem Statement: Let D be a Euclidean ring, F its field of quotients. Prove the Gauss Lemma for polynomials with coefficients in D factored as products of polynomials with coefficients in F.
Gauss' Lemma: If the primitive polynomial f(x) can be factored as the product of two polynomials having rational coefficients, it can be factored as the product of two polynomials having integer coefficients.
The proof of Gauss' Lemma is in the book. However, I'm not even sure how to begin to modify it in order to account for the original coefficients being in D while the factored product's coefficients are in F.
Any help would be greatly appreciated!
Gauss' Lemma: If the primitive polynomial f(x) can be factored as the product of two polynomials having rational coefficients, it can be factored as the product of two polynomials having integer coefficients.
The proof of Gauss' Lemma is in the book. However, I'm not even sure how to begin to modify it in order to account for the original coefficients being in D while the factored product's coefficients are in F.
Any help would be greatly appreciated!