§ Irreducible polynomial over a field divides any polynomial with common root
- Let be an irreducible polynomial over a field . Let it share a common root with another polynomial . Then we claim that divides .
- Consider the GCD . Since share a root , we have that divides . Thus is a non-constant polynomial.
- Further, we have since is GCD. But is irreducible, it cannot be written as product of smaller polynomials, and thus .
- Now, we have , but since , we have . This implies for any that shares a root with .