Category:Definitions/Division over Euclidean Domain

This category contains definitions related to Division over Euclidean Domain.
Related results can be found in Category:Division over Euclidean Domain.

Let $\struct {D, +, \circ}$ be a Euclidean domain:

whose zero is $0_D$

whose Euclidean valuation is denoted $\nu$.

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

By the definition of Euclidean valuation:

$\exists q, r \in D: a = q \circ b + r$

such that either:

$\map \nu r < \map \nu b$

or:

$r = 0_D$

The process of finding $q$ and $r$ is known as division of $a$ by $b$, and we write:

$a \div b = q \rem r$