scholarly journals Minimal degrees of algebraic numbers with respect to primitive elements

2021 ◽  
pp. 1-16
Author(s):  
Cheol-Min Park ◽  
Sun Woo Park
Keyword(s):  
Algebraic Number ◽  
Primitive Element ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Primitive Elements ◽  
Arithmetic Properties ◽  
Congruent Number ◽  
Number Problem ◽  
Number Formula ◽  
Congruent Number Problem

Given a number field [Formula: see text], we define the degree of an algebraic number [Formula: see text] with respect to a choice of a primitive element of [Formula: see text]. We propose the question of computing the minimal degrees of algebraic numbers in [Formula: see text], and examine these values in degree 4 Galois extensions over [Formula: see text] and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves.

scholarly journals Right triangles with algebraic sides and elliptic curves over number fields

2009 ◽  
Vol 59 (3) ◽  
Author(s):  
E. Girondo ◽  
G. González-Diez ◽  
E. González-Jiménez ◽  
R. Steuding ◽  
J. Steuding
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Infinite Order ◽  
Number Fields ◽  
Explicit Construction ◽  
Congruent Number ◽  
Number Problem ◽  
Weil Group ◽  
Congruent Number Problem

AbstractGiven any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P λ of infinite order in the Mordell-Weil group of the elliptic curve Y 2 = X 3 − n 2 X over ℚ(λ).

On the congruent number problem over integers of cyclic number fields

2016 ◽  
Vol 66 (3) ◽  
Author(s):  
Albertas Zinevičius
Keyword(s):  
Number Fields ◽  
Field Extension ◽  
Congruent Number ◽  
Number Problem ◽  
Congruent Number Problem

AbstractGiven a cyclic field extension

scholarly journals On Congruent Numbers

2021 ◽  
Author(s):  
Alex Nguhi
Keyword(s):  
Pythagorean Triples ◽  
Congruent Number ◽  
Number Problem ◽  
Congruent Number Problem

With respect to some classification of Pythagorean triples, if anumber 𝑘 is congruent then it can easily be proven. This expandsthe quest to resolve the congruent number problem. A proposi-tion is put forward on rational sides forming a congruent number.

scholarly journals The congruent number problem

2021 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Finite Number ◽  
Natural Number ◽  
Mathematical Problem ◽  
Straightforward Manner ◽  
Congruent Number ◽  
Number Problem ◽  
Congruent Number Problem ◽  
2D Array

The congruent number problem is the oldest unsolved major mathematical problem to date. The problem aiming to determine whether or not some given integer n is congruent, which corresponds to a Pythagorean triangle with integer sides, can be settled in a finite number of steps. However, once we permit the triangles to acquire rational values for its sides, the degree of difficulty of the task changes dramatically. In this paper a basis is developed, to produce right Pythagorean triangles with rational sides and integral area in a straightforward manner. Determining whether or not a given natural number n is congruent, is equivalent to a search through an ordered 2D array.

scholarly journals Application of the Strong Artin Conjecture to the Class Number Problem

2013 ◽  
Vol 65 (6) ◽  
pp. 1201-1216 ◽  
Author(s):  
Peter J. Cho ◽  
Henry H. Kim
Keyword(s):  
Class Number ◽  
Number Fields ◽  
Class Numbers ◽  
Density Result ◽  
Class Number Formula ◽  
Number Problem ◽  
Number Formula ◽  
Group A ◽  
Galois Closures ◽  
Extremal Value

AbstractWe construct unconditionally several families of number fields with the largest possible class numbers. They are number fields of degree 4 and 5 whose Galois closures have the Galois group A4; S4, and S5. We first construct families of number fields with smallest regulators, and by using the strong Artin conjecture and applying the zero density result of Kowalski–Michel, we choose subfamilies of L-functions that are zero-free close to 1. For these subfamilies, the L-functions have the extremal value at s = 1, and by the class number formula, we obtain the extreme class numbers.

scholarly journals A GENERALIZATION OF THE CONGRUENT NUMBER PROBLEM

2011 ◽  
Vol 07 (08) ◽  
pp. 2237-2247 ◽  
Author(s):  
LARRY ROLEN
Keyword(s):  
Elliptic Curve ◽  
Related Problem ◽  
Fixed Angle ◽  
Root Numbers ◽  
Tate Conjecture ◽  
Congruent Number ◽  
Number Problem ◽  
Parity Conjecture ◽  
Congruent Number Problem

We study a certain generalization of the classical Congruent Number Problem. Specifically, we study integer areas of rational triangles with an arbitrary fixed angle θ. These numbers are called θ-congruent. We give an elliptic curve criterion for determining whether a given integer n is θ-congruent. We then consider the "density" of integers n which are θ-congruent, as well as the related problem giving the "density" of angles θ for which a fixed n is congruent. Assuming the Shafarevich–Tate conjecture, we prove that both proportions are at least 50% in the limit. To obtain our result we use the recently proven p-parity conjecture due to Monsky and the Dokchitsers as well as a theorem of Helfgott on average root numbers in algebraic families.

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