Definition:Coprime/Euclidean Domain
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 $D$.
Notation
Let $a$ and $b$ be objects which in some context are coprime, that is, such that $\gcd \set {a, b} = 1$.
Then the notation $a \perp b$ is preferred on $\mathsf{Pr} \infty \mathsf{fWiki}$.
If $\gcd \set {a, b} \ne 1$, the notation $a \not \!\! \mathop \perp b$ can be used.
Also denoted as
The notation $\perp$ is not universal.
Other notations to indicate the concept of coprimality include:
- $\gcd \set {a, b} = 1$
- $\map \gcd {a, b} = 1$
- $\tuple {a, b} = 1$
However, the first two are unwieldy and the third notation $\tuple {a, b}$ is overused.
Hence the decision by $\mathsf{Pr} \infty \mathsf{fWiki}$ to use $\perp$.
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