Definition:Euclidean Domain/Valuation

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Definition

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 $D$.


Also known as

Some sources refer to a Euclidean valuation as a Euclidean function.

Some sources use the term gauge.


Also see

  • Results about Euclidean valuations can be found here.


Source of Name

This entry was named for Euclid.


Historical Note

A Euclidean valuation is so named because it is central to the definition of a Euclidean domain.

In turn, a Euclidean domain is so named because, as an algebraic structure, it sustains the concept of the Euclidean Algorithm.

Euclid himself made no mention of the concept, which is an abstract algebraic concept defined in (mathematically speaking) modern times.


Linguistic Note

The term Euclidean valuation is introduced with the indefinite article: a Euclidean valuation.

This is because Euclid is pronounced Yoo-klid in English.


Sources