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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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References:
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