Index form equations in sextic fields: a hard computation

Bilu, Yuri; Gaál, István; Győry, Kálmán

doi:10.4064/aa115-1-7

Cited by 35 publications

(27 citation statements)

References 7 publications

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“…This problem has been widely studied and of interest to several mathematicians (cf. [1], [2], [4], [5], [6], [7], [8], [10], [11]). Let K be an algebraic number field generated by a complex root θ of a monic irreducible polynomial f (x) having degree n with coefficients from the ring Z of integers.…”

Section: Introductionmentioning

confidence: 99%

On nonmonogenic number fields defined by

Jakhar¹,

Kumar²

2021

Can. Math. Bull.

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Let q be a prime number and K = Q(θ) be an algebraic number field with θ a root of an irreducible trinomial x 6 +ax+b having integer coefficients. In this paper, we provide some explicit conditions on a, b for which K is not monogenic. As an application, in a special case when a = 0, K is not monogenic if b ≡ 7 mod 8 or b ≡ 8 mod 9. As an example, we also give a non-monogenic class of number fields defined by irreducible sextic trinomials.Throughout the paper, Z K denotes the ring of algebraic integers of an algebraic number field K. For a prime number q and a non-zero m belonging to the ring Z q of q-adic integers, v q (m) will be defined to be the highest power of q dividing m.

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Section: Introductionmentioning

confidence: 99%

On nonmonogenic number fields defined by

Jakhar¹,

Kumar²

2021

Can. Math. Bull.

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“…For results concerning the general cubic case see [9], for the quartic case [8] and for the quintic case [7]. A long computation of about one month on a computer with six parallel processors of 1GH under linux made possible for Bilu, Gaál and Győry to solve an index form equation over a sextic field with Galois group S 6 (see [3]). Further, there are algorithms for solving index form equations also when α has degree ≥ 6 in the case when the field K := Q(α) has proper subfields, or when the Galois group of K is cyclic.…”

Section: An Applicationmentioning

confidence: 99%

An application of index forms in cryptography

Bérczes

Járási

2009

Period Math Hung

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“…There exist general algorithms for solving index form equations in cubic, quartic, quintic, sextic fields, however the general algorithms for quintic and sextic fields are already quite tedious, see [2]. Therefore it is worthy to develop efficient methods for the resolution of special types of higher degree number fields.…”

Section: Introductionmentioning

confidence: 99%

Monogenity in Totally Complex Sextic Fields, Revisited

Gaál¹

2020

JPANTA

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In addition to rather complicated general methods it is interesting and valuable to develop fast efficient methods for calculating generators of power integral bases in special types of number fields. We consider sextic fields containing a real cubic and a complex quadratic fields. We develop a very simple and very efficient method to calculate generators of power integral bases in this type of fields. Our method can be applied to infinite families of number fields, as well. We substantially improve the former methods. Our algorithm is illustrated with detailed examples, involving infinite parametric families.

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Index form equations in sextic fields: a hard computation

Cited by 35 publications

References 7 publications

On nonmonogenic number fields defined by

On nonmonogenic number fields defined by

An application of index forms in cryptography

Monogenity in Totally Complex Sextic Fields, Revisited

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