# 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 $\beta \ne 0$, there exist $\varkappa $ and $\varrho $ of the field such that

$$ |

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

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

where $\varkappa $ is an of the field and $$

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

$$ |

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

$$ |

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 $m\phantom{\rule{veryverythickmathspace}{0ex}}(\ne 0)$ such that $m\gamma $ is an algebraic integer^{} of $K$. The assumption^{} guarantees the integers $\varkappa $, $\varrho $ of $K$ such that

$$ |

Thus

$$ |

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

$$ |

The ends to the remainder 0, because the numbers $|\text{N}({\varrho}_{i})|$ 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

$$d\in \{-11,-7,-3,-2,-1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\hspace{0.17em}5},\mathrm{\hspace{0.17em}6},\mathrm{\hspace{0.17em}7},\mathrm{\hspace{0.17em}11},\mathrm{\hspace{0.17em}13},\mathrm{\hspace{0.17em}17},\mathrm{\hspace{0.17em}19},\mathrm{\hspace{0.17em}21},\mathrm{\hspace{0.17em}29},\mathrm{\hspace{0.17em}33},\mathrm{\hspace{0.17em}37},\mathrm{\hspace{0.17em}41},\mathrm{\hspace{0.17em}57},\mathrm{\hspace{0.17em}73}\}.$$ |

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 |