Definition:Division over Euclidean Domain/Remainder
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Definition
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 $b \ne 0_D$.
Let $q$ and $r$ be the result of division of $a$ by $b$:
- $a = q \circ b + r$ where either $\map \nu r < \map \nu b$ or $r = 0_D$.
Then:
- $r$ is the remainder of the division of $a$ by $b$.
Also see
- Results about remainders of division over a Euclidean domain can be found here.
Sources
- This article incorporates material from Euclidean valuation on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.