Category:Euclidean Valuations

This category contains results about Euclidean Valuations.
Definitions specific to this category can be found in Definitions/Euclidean Valuations.

Let $\struct {D, +, \circ}$ be an integral domain with zero $0_D$.

Let there exist a mapping $\nu: D \setminus \set {0_D} \to \N$ such that for all $a \in D, b \in D_{\ne 0_D}$:

$(1)$	$:$			$\ds \exists q, r \in D: \map \nu r < \map \nu b \text { or } r = 0_D:$	$\ds a = q \circ b + r $
$(2)$	$:$				$\ds \map \nu a \le \map \nu {a \circ b} $

Then $\nu$ is a Euclidean valuation on $D$.

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\((1)\)	$:$			\(\ds \exists q, r \in D: \map \nu r < \map \nu b \text { or } r = 0_D:\)	\(\ds a = q \circ b + r \)
\((2)\)	$:$				\(\ds \map \nu a \le \map \nu {a \circ b} \)

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