Category:Euclidean Valuations
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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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