# Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields

• Volume: 64, Issue: 2, page 465-475
• ISSN: 0011-4642

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## Abstract

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It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial ${P}_{t}\left(x\right)={x}^{4}-t{x}^{3}-6{x}^{2}+tx+1$, assuming that $t>0$, $t\ne 3$ and ${t}^{2}+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.

## How to cite

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Gaál, István, and Petrányi, Gábor. "Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields." Czechoslovak Mathematical Journal 64.2 (2014): 465-475. <http://eudml.org/doc/261976>.

@article{Gaál2014,
abstract = {It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\ne 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.},
author = {Gaál, István, Petrányi, Gábor},
journal = {Czechoslovak Mathematical Journal},
keywords = {simplest quartic field; power integral base; monogeneity; simplest quartic field; power integral base; monogeneity},
language = {eng},
number = {2},
pages = {465-475},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields},
url = {http://eudml.org/doc/261976},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Gaál, István
AU - Petrányi, Gábor
TI - Calculating all elements of minimal index in the infinite parametric family of simplest quartic fields
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 465
EP - 475
AB - It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric family of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $\xi$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\ne 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in all fields in this family.
LA - eng
KW - simplest quartic field; power integral base; monogeneity; simplest quartic field; power integral base; monogeneity
UR - http://eudml.org/doc/261976
ER -

## References

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