Definition:Greatest Common Divisor/Polynomial Ring over Field
Definition
Let $F$ be a field.
Let $P, Q, R \in F \sqbrk X$ be polynomials.
Then $R$ is the greatest common divisor of $P$ and $Q$ if and only if it is a monic greatest common divisor.
This is denoted $\gcd \set {P, Q} = R$.
Also known as
The greatest common divisor is often seen abbreviated as GCD, gcd or g.c.d.
Some sources write $\gcd \set {a, b}$ as $\tuple {a, b}$, but this notation can cause confusion with ordered pairs.
The notation $\map \gcd {a, b}$ is frequently seen, but the set notation, although a little more cumbersome, can be argued to be preferable.
The greatest common divisor is also known as the highest common factor, or greatest common factor.
Highest common factor when it occurs, is usually abbreviated as HCF, hcf or h.c.f.
It is written $\hcf \set {a, b}$ or $\map \hcf {a, b}$.
The archaic term greatest common measure can also be found, mainly in such as Euclid's The Elements.