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 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.

Waring's problem in algebraic number fields

1961 ◽  
Vol 57 (3) ◽  
pp. 449-459 ◽  
Author(s):  
B. J. Birch
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Algebraic Number ◽  
Good Deal ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Singular Series ◽  
Algebraic Number Fields ◽  
Extra Information ◽  
Totally Positive

Let K be a finite algebraic number field, of degree R. Then those integers of K which may be expressed as a sum of dth powers generate a subring JK, d of the integers of K (JK, d need not be an ideal of K, as the simplest example K = Q(i), d = 2 shows. JK, d is an order, it in fact contains all integer multiples of d!; it also contains all rational integers). Siegel(12) showed that every sufficiently large totally positive integer of JK, d is the sum of at most (2d−1 + R) Rd totally positive dth powers; and he conjectured that the number of dth powers necessary should be independent of the field K—for instance, he had proved(11) that five squares are enough for every K. In this paper, we will show that, as far as the analytic part of the argument is concerned, Siegel's conjecture is correct. I have not been able to deal properly with the problem of proving that the singular series is positive; but since Siegel wrote, a good deal of extra information about singular series has been obtained, in particular by Stemmler(14) and Gray(4). The most spectacular consequence of all this is that if p is prime, then every large enough totally positive integer of JK, p is a sum of (2p + 1) totally positive pth. powers.

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.

scholarly journals The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Abdulaziz Deajim
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Relevant Literature ◽  
Number Fields ◽  
Quadratic Number ◽  
Hecke Group ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Imaginary Number

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ ,   c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples are also given.

scholarly journals On the 2-rank and 4-rank of the class group of some real pure quartic number fields

2021 ◽  
Vol 71 (2) ◽  
pp. 285-300
Author(s):  
Mbarek Haynou ◽  
Mohammed Taous
Keyword(s):  
Number Field ◽  
Class Group ◽  
Number Fields ◽  
Prime Divisors ◽  
Prime Integer ◽  
Number Of Prime Divisors ◽  
Real Quadratic

Abstract Let K = ℚ ( p d 2 4 ) $\begin{array}{} \displaystyle (\sqrt[4]{pd^{2}}) \end{array}$ be a real pure quartic number field and k = ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ) its real quadratic subfield, where p ≡ 5 (mod 8) is a prime integer and d an odd square-free integer coprime to p. In this work, we calculate r 2(K), the 2-rank of the class group of K, in terms of the number of prime divisors of d that decompose or remain inert in ℚ( p $\begin{array}{} \displaystyle \sqrt{p} \end{array}$ ), then we will deduce forms of d satisfying r 2(K) = 2. In the last case, the 4-rank of the class group of K is given too.

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.

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.

Number fields with large minimal index containing quadratic subfields

2018 ◽  
Vol 14 (09) ◽  
pp. 2333-2342 ◽  
Author(s):  
Henry H. Kim ◽  
Zack Wolske
Keyword(s):  
Number Field ◽  
Number Fields ◽  
Upper Bounds ◽  
Minimal Index

In this paper, we consider number fields containing quadratic subfields with minimal index that is large relative to the discriminant of the number field. We give new upper bounds on the minimal index, and construct families with the largest possible minimal index.

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