scholarly journals Roots of Elliptic Scator Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 321
Author(s):  
Manuel Fernandez-Guasti
Keyword(s):  
Abelian Group ◽  
Geometric Interpretation ◽  
Roots Of Unity ◽  
Complex Roots ◽  
De Moivre’S Formula

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

COMPOSITION SERIES OF THE SOLVABLE ABELIAN FACTOR GROUP SOURCE OF EQUATION OF ALL POLYNOMIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 462-469
Author(s):  
Bernard Alechenu ◽  
Babayo Muhammed Abdullahi ◽  
Daniel Eneojo Emmanuel
Keyword(s):  
Abelian Group ◽  
Complex Analysis ◽  
Factor Group ◽  
Roots Of Unity ◽  
Composition Series ◽  
Isomorphism Theorem ◽  
Polynomial Equations ◽  
Canonical Map ◽  
The Third ◽  
Recursive Set

This work penciled down the Composition Series of Factor Abelian Group over the source of all polynomial equations gleaned through  the nth roots of unity regular gons on a unit circle, a circle of radius one and centered at zero. To get the composition series, the third isomorphism theorem has to be passed through. But, the third isomorphism theorem itself gleaned via the first which is a deduction of the naturally existing canonical map. The solution of the source atom of the equation of all equation of polynomials are solvable by the intertwine of the Euler’s Formula and the De Moivre’s Theorem which after the inter-math, they become within the domain of complex analysis. For the source root of the equations, there is a recursive set of homomorphisms and ontoness of the mappings geneting the sequential terms in the composition series.    

scholarly journals On theory of finite subsets monoid for one torsion abelian group

2021 ◽  
pp. 39-55
Author(s):  
Сергей Михайлович Дудаков
Keyword(s):  
Abelian Group ◽  
Multiplicative Group ◽  
Torsion Group ◽  
Roots Of Unity

Ранее был доказан следующий результат: если абелева группа $\gG$ не является группой кручения, то теория моноида ее конечных подмножеств позволяет интерпретировать элементарную арифметику. В настоящей работе мы приводим пример, который показывает, что аналогичный результат можно получить и, по крайней мере, для некоторых групп кручения. Earlier it was proved the following claim. Let $\gG$ be a non-torsion abelian group and $\gG$ be the semigroup of finite subsets of $\gG$. Then elementary arithmetic can be interpreted in $\gG^*$, so the theory of $\gG^*$ is undecidable. Here we prove the same result for one torsion group, the multiplicative group of all roots of unity.

scholarly journals Powers of Elliptic Scator Numbers

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 250
Author(s):  
Manuel Fernandez-Guasti
Keyword(s):  
Rational Number ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Geometric Interpretation ◽  
Two Dimensional ◽  
Closed Curves ◽  
Complex Algebra ◽  
Polar Representation ◽  
De Moivre’S Formula ◽  
Three Dimensional Space

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

scholarly journals Generators and Relations for Cyclotomic Units

1966 ◽  
Vol 27 (2) ◽  
pp. 401-407 ◽  
Author(s):  
Hyman Bass
Keyword(s):  
Number Theory ◽  
Roots Of Unity ◽  
Generators And Relations ◽  
Complex Roots ◽  
Complete Set ◽  
Cyclotomic Units

We prove here an unpublished conjecture of Milnor which gives a complete set of multiplicative relations between the numbers e′(ζ) = 1−ζ,where ranges over complex roots of unity. Information of this type is useful in certain areas of topology as well as in number theory.

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