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[parent] norm-Euclidean number field (Topic)

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

$\displaystyle \alpha = \varkappa\beta+\varrho,\;\; \vert$N$\displaystyle (\varrho)\vert < \vert$N$\displaystyle (\beta)\vert.$
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 expressible in the form

$\displaystyle \gamma = \varkappa+\delta$ (1)

where $ \varkappa$ is an integer of the field and $ \vert$N$ (\delta)\vert < 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

$\displaystyle \frac{\alpha}{\beta} = \varkappa+\delta, \quad \vert$N$\displaystyle (\delta)\vert < 1.$
Thus we have
$\displaystyle \alpha = \varkappa\beta+\beta\delta = \varkappa\beta+\varrho.$
Here $ \varrho = \beta\delta$ is integer, since $ \alpha$ and $ \varkappa\beta$ are integers. We also have
$\displaystyle \vert$N$\displaystyle (\varrho)\vert = \vert$N$\displaystyle (\beta)\vert\cdot\vert$N$\displaystyle (\delta)\vert < \vert$N$\displaystyle (\beta)\vert\cdot1 = \vert$N$\displaystyle (\beta)\vert.$
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 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
$\displaystyle m\gamma = \varkappa m+\varrho,$   N$\displaystyle (\varrho) <$   N$\displaystyle (m).$
Thus
$\displaystyle \gamma = \frac{m\gamma}{m} = \varkappa+\frac{\varrho}{m}, \quad \... ...ht)\right\vert = \frac{\vert\mbox{N}(\varrho)\vert}{\vert\mbox{N}(m)\vert} < 1,$
Q.E.D.

Theorem 2. In a norm-Euclidean number field, any two non-zero integers 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{displaymath} \begin{cases} \varrho_0 = \varkappa_2\varrho_1+\varrho_2, \q... ...vert\ \varrho_{n-1} = \varkappa_{n+1}\varrho_n+0, \end{cases}\end{displaymath}
The chain ends to the remainder 0, because the numbers $ \vert$N$ (\varrho_i)\vert$ 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 are norm-Euclidean, e.g. $ \mathbb{Q}(\sqrt{14})$.

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

$\displaystyle d\in\{-11,\,-7,\,-3,\,-2,\,-1,\,2,\,3,\,5,\,6,\,7,\,11,\,13,\,17,\,19,\,21,\,29,\,33,\,37,\,41,\,57,\,73\}.$



"norm-Euclidean number field" is owned by pahio. [ full author list (2) ]
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See Also: Euclidean valuation, quadratic imaginary norm-Euclidean number fields, list of all imaginary quadratic extensions whose ring of integers is a PID, Euclidean field

Also defines:  norm-Euclidean

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quadratic imaginary norm-Euclidean number fields (Theorem) by pahio
number field that is not norm-Euclidean (Example) by pahio
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Cross-references: quadratic fields, principal ideals, ring, ideals, unique factorization domain, ring of integers, units, infinite, Euclid's algorithm, norm and trace of algebraic number, sequence, remainder, divisible, divisor, commutative ring, greatest common divisor, algebraic integer, rational integer, proof, number, function, norm, field, algebraic number field
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This is version 13 of norm-Euclidean number field, born on 2007-03-27, modified 2008-08-08.
Object id is 9124, canonical name is EuclideanNumberField.
Accessed 1483 times total.

Classification:
AMS MSC: 11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
 11R21 (Number theory :: Algebraic number theory: global fields :: Other number fields)
 13F07 (Commutative rings and algebras :: Arithmetic rings and other special rings :: Euclidean rings and generalizations)
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