**Problem**. Let be a field and suppose that are three polynomials in Prove that if both and are irreducible and then

**Solution**. Let and be the ideals of generated by and respectively. Let and Since both and are irreducible, and are field extensions of Now, define the map by for all We first show that is well-defined. To see this, suppose that Then for some and hence because So is well-defined. Now is clearly a ring homomorphism and, since is a field and is an ideal of we must have Therefore we may assume that and hence

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