norm-Euclidean number field

Definition. An algebraic number field $K$ is a norm-Euclidean number field, if for every pair $(\alpha,\,\beta)$ of the integers (http://planetmath.org/AlgebraicInteger) of $K$ , where $\beta\neq 0$ , there exist $\varkappa$ and $\varrho$ of the field such that

$\alpha\;=\;\varkappa\beta+\varrho,\quad|\mbox{N}(\varrho)|<|\mbox{N}(\beta)|.$

Here N means the norm function in $K$ .

Theorem 1. A field $K$ is norm-Euclidean if and only if each number $\gamma$ of $K$ is in the form

$\displaystyle\gamma\;=\;\varkappa+\delta$

(1)

where $\varkappa$ is an of the field and $|\mbox{N}(\delta)|<1.$

Proof. First assume the condition (1). Let $\alpha$ and $\beta$ be integers of $K$ , $\beta\neq 0$ . Then there are the numbers $\varkappa,\,\delta\in K$ such that $\varkappa$ is integer and

$\frac{\alpha}{\beta}\;=\;\varkappa+\delta,\quad|\mbox{N}(\delta)|\;<\;1.$

Thus we have

$\alpha\;=\;\varkappa\beta+\beta\delta\;=\;\varkappa\beta+\varrho.$

Here $\varrho=\beta\delta$ is integer, since $\alpha$ and $\varkappa\beta$ are integers. We also have

$|\mbox{N}(\varrho)|\;=\;|\mbox{N}(\beta)|\cdot|\mbox{N}(\delta)|\;<\;|\mbox{N}% (\beta)|\cdot 1\;=\;|\mbox{N}(\beta)|.$

Accordingly, $K$ is a norm-Euclidean number field. Secondly assume that $K$ is norm-Euclidean. Let $\gamma$ be an arbitrary element of the field. We can determine (http://planetmath.org/MultiplesOfAnAlgebraicNumber) a rational integer $m\,(\neq 0)$ such that $m\gamma$ is an algebraic integer of $K$ . The assumption guarantees the integers $\varkappa$ , $\varrho$ of $K$ such that

$m\gamma\;=\;\varkappa m+\varrho,\quad\mbox{N}(\varrho)\;<\;\mbox{N}(m).$

Thus

$\gamma=\frac{m\gamma}{m}=\varkappa+\frac{\varrho}{m},\quad\left|\mbox{N}\left(% \frac{\varrho}{m}\right)\right|=\frac{|\mbox{N}(\varrho)|}{|\mbox{N}(m)|}<1,$

Q.E.D.

Theorem 2. In a norm-Euclidean number field, any two non-zero have a greatest common divisor.

Proof. We recall that the greatest common divisor of two elements of a commutative ring means such a common divisor of the elements that it is divisible by each common divisor of the elements. Let now $\varrho_{0}$ and $\varrho_{1}$ be two algebraic integers of a norm-Euclidean number field $K$ . According the definition there are the integers $\varkappa_{i}$ and $\varrho_{i}$ of $K$ such that

$\begin{cases}\varrho_{0}=\varkappa_{2}\varrho_{1}+\varrho_{2},\quad|\mbox{N}(% \varrho_{2})|<|\mbox{N}(\varrho_{1})|\\ \varrho_{1}=\varkappa_{3}\varrho_{2}+\varrho_{3},\quad|\mbox{N}(\varrho_{3})|<% |\mbox{N}(\varrho_{2})|\\ \varrho_{2}=\varkappa_{4}\varrho_{3}+\varrho_{4},\quad|\mbox{N}(\varrho_{4})|<% |\mbox{N}(\varrho_{3})|\\ \qquad\cdots\cdots\\ \varrho_{n-2}=\varkappa_{n}\varrho_{n-1}+\varrho_{n},\;\;|\mbox{N}(\varrho_{n}% )|<|\mbox{N}(\varrho_{n-1})|\\ \varrho_{n-1}=\varkappa_{n+1}\varrho_{n}+0,\end{cases}$

The ends to the remainder 0, because the numbers $|\mbox{N}(\varrho_{i})|$ form a descending sequence of non-negative rational integers — see the entry norm and trace of algebraic number. As in the Euclid’s algorithm in $\mathbb{Z}$ , one sees that the last divisor $\varrho_{n}$ is one greatest common divisor of $\varrho_{0}$ and $\varrho_{1}$ . N.B. that $\varrho_{0}$ and $\varrho_{1}$ may have an infinite amount of their greatest common divisors, depending the amount of the units in $K$ .

Remark. The ring of integers of any norm-Euclidean number field is a unique factorization domain and thus all ideals of the ring are principal ideals. But not all algebraic number fields with ring of integers a UFD (http://planetmath.org/UFD) are norm-Euclidean, e.g. $\mathbb{Q}(\sqrt{14})$ .

Theorem 3. The only norm-Euclidean quadratic fields $\mathbb{Q}(\sqrt{d})$ are those with

$d\in\{-11,\,-7,\,-3,\,-2,\,-1,\,2,\,3,\,5,\,6,\,7,\,11,\,13,\,17,\,19,\,21,\,2% 9,\,33,\,37,\,41,\,57,\,73\}.$

Title	norm-Euclidean number field
Canonical name	NormEuclideanNumberField
Date of creation	2013-03-22 16:52:26
Last modified on	2013-03-22 16:52:26
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	17
Author	pahio (2872)
Entry type	Topic
Classification	msc 13F07
Classification	msc 11R21
Classification	msc 11R04
Related topic	EuclideanValuation
Related topic	QuadraticImaginaryEuclideanNumberFields
Related topic	ListOfAllImaginaryQuadraticPIDs
Related topic	EuclideanField
Related topic	AlgebraicNumberTheory
Related topic	MixedFraction
Defines	norm-Euclidean