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A family of cyclic quartic fields with explicit fundamental units. (English) Zbl 1442.11141

In this paper, the authors prove a new result about the fundamental units of a family of cyclic quartic fields :
Let \(s\) be an integer such that \(3s^2-4s+4\) is a square. Let \(K_s\) be the splitting field of \[F_s(t) = t^4 + (4s^3-4s^2 + 8s - 4)t^3 + (-6s^2 - 6)t^2 + 4t + 1.\] Then \(\text{Gal}(K_s/\mathbb Q)\) is cyclic of order \(4\). If \(s^2 + 2\) is squarefree and \(s\neq0\), then \(\pm1\) and the roots of \(F_s(t)\) generate either the unit group of the ring of algebraic integers in \(K_s\) or a subgroup of index \(5\).

MSC:

11R16 Cubic and quartic extensions
11R27 Units and factorization

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References:

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