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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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