Integral bases and monogenity of the simplest sextic fields

Gaál, István; Remete, László

doi:10.4064/aa170502-23-10

Cited by 6 publications

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“…Similar phenomenon was proved for simplest sextic fields, if m 2 + 3m + 9 is square-free (see I. Gaál and L. Remete [6]).…”

Section: Introductionsupporting

confidence: 73%

Integral bases of pure fields with square-free parameter

Remete

2020

Studia Scientiarum Mathematicarum Hungarica

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Let m ≠ 0, ±1 and n ≥ 2 be integers. The ring of algebraic integers of the pure fields of type is explicitly known for n = 2, 3,4. It is well known that for n = 2, an integral basis of the pure quadratic fields can be given parametrically, by using the remainder of the square-free part of m modulo 4. Such characterisation of an integral basis also exists for cubic and quartic pure fields, but for higher degree pure fields there are only results for special cases. In this paper we explicitly give an integral basis of the field , where m ≠ ±1 is square-free. Furthermore, we show that similarly to the quadratic case, an integral basis of is repeating periodically in m with period length depending on n.

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“…Similar phenomenon was proved for simplest sextic fields, if m 2 + 3m + 9 is square-free (see I. Gaál and L. Remete [6]).…”

Section: Introductionsupporting

confidence: 73%

Integral bases of pure fields with square-free parameter

Remete

2020

Studia Scientiarum Mathematicarum Hungarica

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“…These fields contained the simplest cubic fields L m defined by Shanks [24]. Recently, Gaal [9] found an explicit integral basis of S m and proved that S m are not monogenic except for a few values of m = −4, −2, −1, 1.…”

Section: Resultsmentioning

confidence: 99%

On the indices in number fields and their computation for small degrees

Bayad

Seddik

2021

APPL ANAL DISCRETE M

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Let K be a number field. We investigate the indices I(K) and i(K) of K introduced respectively by Dedekind and Gunji-McQuillan. Let n be a positif integer, we then prove that for any prime p ? n, there exists K a number field of degree n over Q such that p divide i(K). This result is an analogue to Bauer?s one for i(K). We compute I(K) and i(K) for cubic fields and infinite families of simplest number fields of degree less than 7. We solve questions and disprove the conjecture stated in.

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“…✷ Theorem 10. If K admits a power integral basis, then the following divisibility conditions must hold: n mod 8 m mod 16 1 2 3 4 5 6 1, 5 1,3,5,7,11,13,15 n | m 2 3,5,7,9,11,13,15 4n If the right hand term in the divisibility conditions of columns 5 and 6 does not reduce to 0, there remain only a few possible values for m 0 . Proof of Theorem 10.…”

Section: Composites Of Quadratic Fields and The Simplest Quartic Fieldsmentioning

confidence: 99%

“…In some recent papers [9], [11] the authors developed a new and efficient technics to consider monogenity in infinite parametric families of higher degree number fields. The most important features of this method are that -the integral bases are determined in a parametric form, -the factors of the index form are explicitly calculated, -some linear combinations of these factors are shown to have some non-trivial divisors.…”

Section: Introductionmentioning

confidence: 99%

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Integral Bases and Monogenity of Composite Fields

Gaál

Remete

2017

Experimental Mathematics

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We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an integral basis of the composite fields. We construct the index form, describe their factors and prove that the monogenity of the composite fields imply certain divisibility conditions on the parameters involved. These conditions usually can not hold, which implies the non-monogenity of the fields.The fields that we consider are higher degree number fields, of degrees 6 up to 12. The non-monogenity of the number fields is stated very often as a consequence of the non-existence of the solutions of the index form equation. Up to our knowlegde it is not at all feasible to solve the index form equation in these high degree fields, especially not in a parametric form.On the other hand our method implies directly the non-monogenity in almost all cases. We obtain our results in a parametric form, characterizing these infinite parametric families of composite fields.

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Integral bases and monogenity of the simplest sextic fields

Cited by 6 publications

References 0 publications

Integral bases of pure fields with square-free parameter

Integral bases of pure fields with square-free parameter

On the indices in number fields and their computation for small degrees

Integral Bases and Monogenity of Composite Fields

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