Definition:Euclidean Domain
Definition
Let $\struct {D, +, \circ}$ be an integral domain.
Let there exist a Euclidean valuation on $D$.
Then $D$ is called a Euclidean domain.
Euclidean Valuation
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 $D$.
Examples
Integers are Euclidean Domain
The integers $\Z$ with the mapping $\nu: \Z \to \Z$ defined as:
- $\forall x \in \Z: \map \nu x = \size x$
form a Euclidean domain.
Polynomial Forms over Field is Euclidean Domain
Let $\struct {F, +, \circ}$ be a field whose zero is $0_F$ and whose unity is $1_F$.
Let $X$ be transcendental in $F$.
Let $F \sqbrk X$ be the ring of polynomial forms in $X$ over $F$.
Then $F \sqbrk X$ is a Euclidean domain.
Gaussian Integers form Euclidean Domain
Let $\struct {\Z \sqbrk i, +, \times}$ be the integral domain of Gaussian Integers.
Let $\nu: \Z \sqbrk i \to \R$ be the real-valued function defined as:
- $\forall a \in \Z \sqbrk i: \map \nu a = \cmod a^2$
where $\cmod a$ is the (complex) modulus of $a$.
Then $\nu$ is a Euclidean valuation on $\Z \sqbrk i$.
Hence $\struct {\Z \sqbrk i, +, \times}$ with $\nu: \Z \sqbrk i \to \Z$ forms a Euclidean domain.
Examples of Use of Euclidean Algorithm
GCD of $5 i$ and $3 + i$ in Ring of Gaussian Integers
The GCD of $5 i$ and $3 + 1$ in the ring of Gaussian integers is found to be:
- $\gcd \set {5 i, 3 + 1} = 1 + 2 i$
and its associates $-1 - 2 i$, $-2 + i$ and $2 - i$.
Also known as
A Euclidean domain is also known as a Euclidean ring.
Also see
- Results about Euclidean domains can be found here.
Source of Name
This entry was named for Euclid.
Historical Note
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 domain is introduced with the indefinite article: a Euclidean domain.
This is because Euclid is pronounced Yoo-klid in English.
Sources
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $6$: Polynomials and Euclidean Rings: $\S 27$. Euclidean Rings
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Euclidean domain or Euclidean ring