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Generalization of a Shanks’ family. (Généralisation d’une famille de Shanks.) (French) Zbl 0899.11066
The author considers the family of fields $$K= \mathbb{Q}(\alpha)$$, where $$\alpha$$ is a root of $f(x)= x^3- rc^m x^2- (c^t-1)x- rc^m$ with $$c\geq 2$$, $$m\geq 1$$, $$1\leq t\leq m$$ and $$r\mid (c^{(m,t)}- 1)$$. Two cases are dealt with: $$r=1$$ and $$r\neq 1$$. Suppose first $$r\neq 1$$. She takes the Jacobi-Perron algorithm of the vector $$(\alpha (\alpha- rc^m),\alpha)$$ and proves that it is purely periodic of length $$\ell= 3(3m+t)/ (m,t)$$. This provides her with the unit $$\varepsilon= \frac{\alpha^{3t/d}}{r} (\alpha (\alpha- rc^m)^{-1})^{3m/d}$$ in $$\mathbb{Z}[\alpha]$$. She also develops the Voronoï algorithm (again the length is $$3(3m+t)/ (m,t)$$), and shows that the above unit $$\varepsilon$$ is the fundamental unit of $$\mathbb{Z}[\alpha]$$. The same results hold in the case $$r=1$$, except that $$\ell= (3m+t)/ (m,t)$$ and $$\varepsilon= \alpha^{t/d} (\alpha (\alpha- c^m)^{-1})^{m/d}$$. The paper is very well written, but the choice of the title is bad, since the link with a family of quadratic fields of Shanks deserves only a minor remark.
##### MSC:
 11Y65 Continued fraction calculations (number-theoretic aspects) 11A55 Continued fractions 11R27 Units and factorization 11J70 Continued fractions and generalizations
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