Polynomial greatest common divisor

Summary

The greatest common divisor of two polynomials over a field is the monic common divisor that every common divisor divides. It is the greatest common divisor in the ring $K[x]$, made unique by asking it to be monic. The Euclidean algorithm computes it with repeated long division. It detects common roots, and $\gcd(f, f')$ detects repeated roots.