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Fundamental units in a parametric family of not totally real quintic number fields. (English) Zbl 1119.11065

For \(n \in {\mathbb Z}\), we consider the polynomials \(P_{n}(X) = X^{5} -nX^{4} + (2n-1)X^{3} - (n-2)X^{2} - 2X + 1\) which are irreducibles and have signature \((1,2)\) for \(n\leq 6\). Let \(\rho\) be a root of \(P_{n}(X)\) and \(K\) be the a not totally real quintic number field generated by \(\rho\). In this paper, the author proves that \(\rho\) and \(\rho - 1\) are independent units in \({\mathbb Z}[\rho]\) for \(n\leq 6\), and that for \(n < 6\) \(\{\rho, \rho - 1\}\) is a system of fundamental units of \(K\).

MSC:

11R27 Units and factorization
11R21 Other number fields
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References:

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