On Power Integral Bases for Certain Pure Number Fields Defined by x 36 − m

2021 ◽  
Vol 58 (3) ◽  
pp. 371-380
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Number Field ◽  
Irreducible Polynomial ◽  
Number Fields ◽  
Rational Integer ◽  
Complex Root ◽  
Integral Bases ◽  
Pure Number

Let K = ℚ(α) be a number field generated by a complex root a of a monic irreducible polynomial ƒ (x) = x36 − m, with m ≠ ±1 a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≠ ±1 (mod 9) then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡±1 (mod 9), then the number field K is not monogenic.

On power integral bases for certain pure number fields defined by x24 – m

2020 ◽  
Vol 57 (3) ◽  
pp. 397-407
Author(s):  
Lhoussain El Fadil
Keyword(s):  
Number Field ◽  
Irreducible Polynomial ◽  
Number Fields ◽  
Rational Integer ◽  
Complex Root ◽  
Integral Bases ◽  
Pure Number

AbstractLet K = ℚ(α) be a number field generated by a complex root α of a monic irreducible polynomial f(x) = x24 – m, with m ≠ 1 is a square free rational integer. In this paper, we prove that if m ≡ 2 or 3 (mod 4) and m ≢∓1 (mod 9), then the number field K is monogenic. If m ≡ 1 (mod 4) or m ≡ 1 (mod 9), then the number field K is not monogenic.

On monogenity of certain pure number fields defined by xpr − m

2021 ◽  
Author(s):  
Hamid Ben Yakkou ◽  
Lhoussain El Fadil
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Irreducible Polynomial ◽  
Number Fields ◽  
Rational Integer ◽  
Complex Root ◽  
Prime Integer ◽  
Pure Number ◽  
Polynomial Formula

Let [Formula: see text] be a pure number field generated by a complex root [Formula: see text] of a monic irreducible polynomial [Formula: see text] where [Formula: see text] is a square free rational integer, [Formula: see text] is a rational prime integer, and [Formula: see text] is a positive integer. In this paper, we study the monogenity of [Formula: see text]. We prove that if [Formula: see text], then [Formula: see text] is monogenic. But if [Formula: see text] and [Formula: see text], then [Formula: see text] is not monogenic. Some illustrating examples are given.

On the extensions of discrete valuations in number fields

2019 ◽  
Vol 69 (5) ◽  
pp. 1009-1022
Author(s):  
Abdulaziz Deajim ◽  
Lhoussain El Fadil
Keyword(s):  
Number Field ◽  
Irreducible Polynomial ◽  
Number Fields ◽  
Irreducible Polynomials

Abstract Let K be a number field defined by a monic irreducible polynomial F(X) ∈ ℤ [X], p a fixed rational prime, and νp the discrete valuation associated to p. Assume that F(X) factors modulo p into the product of powers of r distinct monic irreducible polynomials. We present in this paper a condition, weaker than the known ones, which guarantees the existence of exactly r valuations of K extending νp. We further specify the ramification indices and residue degrees of these extended valuations in such a way that generalizes the known estimates. Some useful remarks and computational examples are also given to highlight some improvements due to our result.

On power integral bases for certain pure number fields defined by x2r.5s−m

2021 ◽  
pp. 1-11
Author(s):  
Hamid Ben Yakkou ◽  
Abdelhakim Chillali ◽  
Lhoussain El Fadil
Keyword(s):  
Number Fields ◽  
Integral Bases ◽  
Pure Number

scholarly journals Hyperbolic tessellations and generators of for imaginary quadratic fields

2021 ◽  
Vol 9 ◽  
Author(s):  
David Burns ◽  
Rob de Jeu ◽  
Herbert Gangl ◽  
Alexander D. Rahm ◽  
Dan Yasaki
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Quadratic Fields ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Quadratic Number Fields ◽  
Formula Group ◽  
Imaginary Quadratic Number ◽  
Direct Calculations

Abstract We develop methods for constructing explicit generators, modulo torsion, of the $K_3$ -groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic $3$ -space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite $K_3$ -group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for $ K_3 $ of any field, predict the precise power of $2$ that should occur in the Lichtenbaum conjecture at $ -1 $ and prove that this prediction is valid for all abelian number fields.

Power Integral Bases in Composits of Number Fields

1998 ◽  
Vol 41 (2) ◽  
pp. 158-165 ◽  
Author(s):  
István Gaál
Keyword(s):  
Number Fields ◽  
Parametric Family ◽  
Prime Degree ◽  
Index Form ◽  
Form Equation ◽  
Integral Bases ◽  
Quadratic Extensions ◽  
Totally Real

AbstractIn the present paper we consider the problem of finding power integral bases in number fields which are composits of two subfields with coprime discriminants. Especially, we consider imaginary quadratic extensions of totally real cyclic number fields of prime degree. As an example we solve the index form equation completely in a two parametric family of fields of degree 10 of this type.

scholarly journals On the complex dynamics of birational surface maps defined over number fields

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Mattias Jonsson ◽  
Paul Reschke
Keyword(s):  
Number Field ◽  
Complex Dynamics ◽  
Energy Condition ◽  
Height Function ◽  
Number Fields ◽  
Projective Surface ◽  
Complex Projective Surface ◽  
Canonical Height ◽  
Dynamical Degree ◽  
Complex Projective

AbstractWe show that any birational selfmap of a complex projective surface that has dynamical degree greater than one and is defined over a number field automatically satisfies the Bedford–Diller energy condition after a suitable birational conjugacy. As a consequence, the complex dynamics of the map is well behaved. We also show that there is a well-defined canonical height function.

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