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L’anneau $${\mathbb{Z}}[\sqrt{14}]$$ et l’algorithme euclidien. (The ring $${\mathbb{Z}}[\sqrt{14}]$$ and the Euclidean algorithm). (English) Zbl 0576.12003
It is known that the rings of integers of real quadratic fields with class number equal to 1 are Euclidean if certain generalized Riemann hypotheses are true. There are only 16 of those fields for which the norm function is a Euclidean algorithm. The field $${\mathbb{Q}}(\sqrt{14})$$ is a quadratic field with class number equal to 1 for which the norm function is not a Euclidean algorithm.
The author investigates whether it can be proven, without using Riemann hypotheses, that the ring of integers $${\mathbb{Z}}[\sqrt{14}]$$ of $${\mathbb{Q}}(\sqrt{14})$$ has a Euclidean algorithm. Therefore he proceeds in two stages. First he characterizes the pairs of elements of $${\mathbb{Z}}[\sqrt{14}]$$ for which the Euclidean condition for the norm does not hold, i.e. those pairs a,b$$\in {\mathbb{Z}}[\sqrt{14}]$$ such that for all $$q\in {\mathbb{Z}}[\sqrt{14}]$$ we have $$N(a+qb)\geq N(b)$$. Using this information he tries to modify the norm function in such a way that the Euclidean condition holds for these pairs and of course still holds for all other pairs. The author suggest to take as Euclidean function $$\phi (a)=0$$ if $$a=0$$, $$\phi (a)=(3/2)^{[\nu (a)/2]}\cdot N(a)$$ if $$a\neq 0$$, where $$\nu$$ is the 2-adic valuation in $${\mathbb{Q}}(\sqrt{14}).$$
The proof that this function is a Euclidean algorithm relies on a yet unproven assumption. This assumption states that for each odd integer $$z\in {\mathbb{Z}}$$ and for each $$a\in {\mathbb{Z}}[\sqrt{14}]$$, not divisible by z there exists q,r$$\in {\mathbb{Z}}[\sqrt{14}]$$ such that $$a=zq+r$$ and $$N(r)<(2/3)^{[(\nu (a)+1)/2]}\cdot N(z)$$. The evidence for this assumption lies in the fact that for ’almost all’ $$a\in {\mathbb{Z}}[\sqrt{14}]$$ there is a Euclidean remainder mod z of odd norm. Another evidence is that the assumption holds for all odd z with $$| z| <150$$.
Reviewer: F.J.van der Linden

##### MSC:
 11R11 Quadratic extensions 11R04 Algebraic numbers; rings of algebraic integers 11A63 Radix representation; digital problems
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