# Definition:Coprime/Euclidean Domain

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## Definition

Let $\struct {D, +, \times}$ be a Euclidean domain.

Let $U \subseteq D$ be the group of units of $D$.

Let $a, b \in D$ such that $a \ne 0_D$ and $b \ne 0_D$

Let $d = \gcd \set {a, b}$ be the greatest common divisor of $a$ and $b$.

Then $a$ and $b$ are **coprime** if and only if $d \in U$.

That is, two elements of a Euclidean domain are **coprime** if and only if their greatest common divisor is a unit of $D$.

## Also known as

The statement **$a$ and $b$ are coprime** can also be expressed as:

**$a$ and $b$ are relatively prime****$a$ and $b$ are mutually prime****$a$ is prime to $b$**, and at the same time that**$b$ is prime to $a$**.

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 29$. Irreducible elements