Relatively prime polynomials over a field have no common roots

Let F be a field and let p,q \in F[x] be relatively prime. Prove that p and q can have no roots in common.


We noted previously that F[x] is a Bezout domain. In particular, if p and q are relatively prime then there exist a,b \in F[x] such that ap + bq = 1. If \zeta is a root of both p and q, then we have 1 = 0, a contradiction.

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