Power integral bases in a parametric family of totally real cyclic quintics. (English) Zbl 0899.11064

Gaál, István; Pohst, Michael

Math. Comput. 66, No. 220, 1689-1696 (1997).

Authors’ abstract: We consider the totally real cyclic quintic fields \(K_n= \mathbb{Q} (\vartheta_n)\), generated by a root \(\vartheta_n\) of the polynomial \[ \begin{split} f_n(x)= x^5+ n^2\cdot x^4- (2n^3+ 6n^2+ 10n+ 10)x^3+ (n^4+ 5n^3+ 11n^2+ 15n+ 15)x^2\\ +(n^3+ 4n^2+ 10n+ 10)x+ 1. \end{split} \] Assuming \(m= n^4+ 5n^3+ 15n^2+ 25n+25\) is square-free, we compute explicitly an integral basis and a set of fundamental units of \(K_n\) and prove that \(K_n\) has a power integral basis only for \(n=-1,-2\). For \(n=-1,-2\) (both values presenting the same field) all generators of power integral bases are computed.

Reviewer: N.Tzanakis (Iraklion)

MSC:

11Y50	Computer solution of Diophantine equations
11Y40	Algebraic number theory computations
11D57	Multiplicative and norm form equations

Keywords:

family of quintic fields; totally real cyclic quintic fields; integral basis; set of fundamental units; power integral basis

Software:

Maple

References:

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