The monic greatest common divisor of two polynomials over a field

Let $F$ be a field and let $f,g\in F[x]$ be not both zero. Since $F[x]$ is a principal ideal domain (For every field $F$, $F[x]$ is a principal ideal domain), the ideal $(f,g)$ generated by $f$ and $g$ (The ideal generated by a subset and principal ideals) has a nonzero generator $d$. Multiplying $d$ by the inverse of its leading coefficient gives a monic generator, and any two monic generators of the same ideal are equal. The resulting polynomial is the monic greatest common divisor $\gcd(f,g)$.

Equivalently, $d=\gcd(f,g)$ is the unique monic polynomial such that $d$ divides both $f$ and $g$, and every common divisor of $f$ and $g$ divides $d$. The equivalence and the Bézout identity are proved in Bézout identity and the Euclidean algorithm for polynomials over a field ↗. The expression $\gcd(0,0)$ is left undefined.