 Bézout's identity for polynomials

For univariate polynomials $f,g$ over a field, not both zero, there are polynomials $A,B$ such that $Af+Bg=d$, where $d$ is their monic greatest common divisor. In particular, coprime polynomials admit $Af+Bg=1$. The extended Euclidean algorithm constructs the coefficients.