Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields. (English) Zbl 1340.11102

Gaál, István; Petrányi, Gábor

Czech. Math. J. 64, No. 2, 465-475 (2014).

A classical problem in algebraic number theory is to characterize all monogenic fields \(K\), that is fields \(K\) that admit a power integral bases. In other words the classification of number fields \(K\) that admit a solution to the index form equation \[ [\mathbb Z_K:\mathbb Z[\theta]]=1 \] with \(\theta \in \mathbb Z_K\). Therefore the question for the minimal index of \([\mathbb Z_K:\mathbb Z[\theta]]\) seems to be natural. In the paper under review the authors attack this problem in the case that \(K\) lies in the family of simplest quartic fields, that is \(K=\mathbb Q(\xi)\), where \(\xi \) is a root of the polynomial \[ x^4-tx^3-6x^2+tx+1. \] This is achieved if \(t\neq 3\) and \(t^2+16\) is not divisible by an odd square by using a method due to I. Gaál et al. [J. Number Theory 57, No. 1, 90–104 (1996; Zbl 0853.11023)] and a result due to H. K. Kim and J. H. Lee [“Evaluation of the Dedekind zeta functions at \(s=-1\) of the simplest quartic fields”, Trends in Math., New Ser., Inf. Center for Math. Sci. 11, No. 2, 63–79 (2009), http://trends.mathnet.or.kr/mathnet/thesis_file/09-junholee.pdf].

Reviewer: Volker Ziegler (Salzburg)

MSC:

11Y50	Computer solution of Diophantine equations
11D57	Multiplicative and norm form equations
11R04	Algebraic numbers; rings of algebraic integers
11R16	Cubic and quartic extensions

Keywords:

simplest quartic field; power integral base; monogeneity

Citations:

Zbl 0853.11023

Software:

KANT/KASH; Maple

References:

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