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

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].

### 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

Zbl 0853.11023

KANT/KASH; Maple
Full Text:

### References:

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