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Homenumber field that is not norm-Euclidean

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# number field that is not norm-Euclidean

Proposition. The real quadratic field $\mathbb{Q}(\sqrt{14})$ is not norm-Euclidean.

Proof. We take the number $\gamma=\frac{1}{2}+\frac{1}{2}\sqrt{14}$ which is not integer of the field ($14\equiv 2\;\;(\mathop{{\rm mod}}4)$). Antithesis: $\gamma=\varkappa+\delta$ where $\varkappa=a+b\sqrt{14}$ is an integer of the field ($a,\,b\in\mathbb{Z}$) and

$|\mbox{N}(\delta)|=\left|\left(\frac{1}{2}-a\right)^{2}-14\left(\frac{1}{2}-b% \right)^{2}\right|<1.$ |

Thus we would have

$|\underbrace{(2a-1)^{2}-14(2b-1)^{2}}_{{E}}|<4.$ |

And since $(2a-1)^{2}=4(a-1)a+1\equiv 1\;\;(\mathop{{\rm mod}}8)$, it follows $E\equiv 1-14\cdot 1\equiv 3\;\;(\mathop{{\rm mod}}8)$, i.e. $E=3$. So we must have

$\displaystyle(2a-1)^{2}\equiv(2a-1)^{2}-14(2b-1)^{2}\equiv 3\;\;(\mathop{{\rm mod% }}7).$ | (1) |

But $\{0,\,\pm 1,\,\pm 2,\,\pm 3\}$ is a complete residue system modulo 7, giving the set $\{1,\,2,\,4\}$ of possible quadratic residues modulo 7. Therefore (1) is impossible. The antithesis is wrong, whence the theorem 1 of the parent entry says that the number field is not norm-Euclidean.

Note. The function N used in the proof is the usual norm

$\mbox{N}:\,\,r\!+\!s\sqrt{14}\,\mapsto\,r^{2}\!-\!14s^{2}\quad(r,\,s\in\mathbb% {Q})$ |

defined in the field $\mathbb{Q}(\sqrt{14})$. The notion of norm-Euclidean number field is based on the norm. There exists a fainter function, the so-called Euclidean valuation, which can be defined in the maximal orders of some algebraic number fields; such a maximal order, i.e. the ring of integers of the number field, is then a Euclidean domain. The existence of a Euclidean valuation guarantees that the maximal order is a UFD and thus a PID. Recently it has been shown the existence of the Euclidean domain $\mathbb{Z}[\frac{1+\sqrt{69}}{2}]$ in the field $\mathbb{Q}(\sqrt{69})$ but the field is not norm-Euclidean.

The maximal order $\mathbb{Z}[\sqrt{14}]$ of $\mathbb{Q}(\sqrt{14})$ has also been proven to be a Euclidean domain (Malcolm Harper 2004 in Canadian Journal of Mathematics).

## Mathematics Subject Classification

13F07*no label found*11R21

*no label found*11R04

*no label found*

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