# Definition:Greatest Common Divisor/Polynomial Ring over Field

## Definition

Let $F$ be a field.

Let $P, Q, R \in F \left[{X}\right]$ 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 $\gcd \left({P, Q}\right) = R$.

## 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*.