Solubility of algebraic structures is what gleaned the introduction of group theory, which later stems the other realms of abstract algebra viz: rings, fields and semigroup theories. The nth roots of unity is found in the most sensitive texts ever in the history of abstract algebra: Cauchy’s, Galois’ and Cayley’s. These three giant group theorists had the common ground of the roots of unity in even the title of their works. The idea is that if the nth roots of unity are solvable by radicals and so do the composition series approach, then all other products of the nth roots of unity - which the unity itself is part of - will automatically be solvable. Hence, all equations that dissolve to the least of the nth roots of unity are solvable by the composition series. This article penciled down how the congruence modulo of arithmetics due to Gauss and Leibnitz were used to break down the nth roots of unity, so that the recursive process can generate the composition series of normal subgroups between the unity and the group itself. Since they are P-Groups, they have normal P-Sylow Subgroups. The normality comes from the Index Theorem. Because they all have index 2 in their P-Groups, they are the maximal proper normal P-Sylow Subgroups and their factor groups are abelian accounting to the solubility of nth roots of unity by composition series. We combine the classical Euler Formula and the De Moivre Theorem to present the solvability of nth roots of unity. The P-Groups over nth roots of unity are multiplicative. nth roots of unity are subsequences of nth roots of unity and it converges to the limit point of the nth roots of unity.
| Published in | Engineering and Applied Sciences (Volume 6, Issue 3) |
| DOI | 10.11648/j.eas.20210603.12 |
| Page(s) | 49-54 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
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Copyright © The Author(s), 2024. Published by Science Publishing Group |
Cauchy Sequence, Rouche Theorem, Class Equation, Aleph Naught
| [1] | Galois E, Oueveres Mathematiques, Journal de Liouville, Converted to Djvu format by Prof. Antoine Chambert – Loir at the University of Reines, 1846. |
| [2] | Abel N H, Ouevres Completes de Neils Henrik Abel, Norway 1829. |
| [3] | Buya S B, The Bring-Jerrad Quintic Equation, Its Solvability by Factorization into Cubic and Quadratic Factors, Journal of Applied Science and Innovations, 1 (2017): 16-21. |
| [4] | Cayley A, On the Theory of Groups, as Depending on the Symbolic Equation, θn=1 Philosophical Magazine, 4th Series, 7 (42): 40-47. |
| [5] | Weber H, Lehbuch der Algebra, Braunschweig, Robert Fricke, 1924. |
| [6] | Dyck WV, Gruppen theoretische Studien, Methematische Annalen, 20 (1) (1882): 1-44. |
| [7] | Richard L R, A History of Lagrange Theorem on Groups, Mathematics Magazine, 74 (2) (2001): 99 - 108. |
| [8] | Abbati P M, Dizionario Biografico degli Italiani, Vol. I, 1960. |
| [9] | Richard D, Essays on the Theory of Numbers. Open Court Publishing Company, Chicago, 1901. |
| [10] | Heinstein I N, Topics in Algebra, John Wiley & Sons, 2nd Edition, 1975. |
| [11] | John J O’, Robertson E F, Mac Tutor History of Mathematics, University of St Andrews, Scotland, UK, 2017. |
| [12] | Vasistha A R, Vasistha A K, Modern Algebra (Abstract Algebra), Krishna Parakashan Media (P) Ltd, Meerut, Delhi, 2006. |
| [13] | Christopher H, The Early Development of the Algebraic Theory of Semigroups, Archive for History of Exact Sciences, 63 (5) (2009): 497-536. |
| [14] | Stanley B and Sankappanavar H P, A Course in Universal Algebra, Springer-Varlag, United States of America, 1981. |
| [15] | Feit W, Thompson G J, Solvability of Groups of Odd Order, Pacific J. Math, 13 (3) (1963): 775-1029. |
| [16] | Burnside W, Theory of Groups of Finite Order, Cambridge University Press, 1897. |
| [17] | Fraleigh J B, A First Course in Abstract Algebra, Waterstines, 7Ed, 2003. |
| [18] | Seymour L, Murray R S, John J S, Dennis S, Complex Variables, McGraw-Hill Companies, United States of America, 2nd Edition, 2009. |
APA Style
Alechenu Benard, Babayo Muhammed Abdullahi, Daniel Eneojo Emmanuel. (2021). Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach. Engineering and Applied Sciences, 6(3), 49-54. https://doi.org/10.11648/j.eas.20210603.12
ACS Style
Alechenu Benard; Babayo Muhammed Abdullahi; Daniel Eneojo Emmanuel. Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach. Eng. Appl. Sci. 2021, 6(3), 49-54. doi: 10.11648/j.eas.20210603.12
AMA Style
Alechenu Benard, Babayo Muhammed Abdullahi, Daniel Eneojo Emmanuel. Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach. Eng Appl Sci. 2021;6(3):49-54. doi: 10.11648/j.eas.20210603.12
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Download@article{10.11648/j.eas.20210603.12,
author = {Alechenu Benard and Babayo Muhammed Abdullahi and Daniel Eneojo Emmanuel},
title = {Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach},
journal = {Engineering and Applied Sciences},
volume = {6},
number = {3},
pages = {49-54},
doi = {10.11648/j.eas.20210603.12},
url = {https://doi.org/10.11648/j.eas.20210603.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.eas.20210603.12},
abstract = {Solubility of algebraic structures is what gleaned the introduction of group theory, which later stems the other realms of abstract algebra viz: rings, fields and semigroup theories. The nth roots of unity is found in the most sensitive texts ever in the history of abstract algebra: Cauchy’s, Galois’ and Cayley’s. These three giant group theorists had the common ground of the roots of unity in even the title of their works. The idea is that if the nth roots of unity are solvable by radicals and so do the composition series approach, then all other products of the nth roots of unity - which the unity itself is part of - will automatically be solvable. Hence, all equations that dissolve to the least of the nth roots of unity are solvable by the composition series. This article penciled down how the congruence modulo of arithmetics due to Gauss and Leibnitz were used to break down the nth roots of unity, so that the recursive process can generate the composition series of normal subgroups between the unity and the group itself. Since they are P-Groups, they have normal P-Sylow Subgroups. The normality comes from the Index Theorem. Because they all have index 2 in their P-Groups, they are the maximal proper normal P-Sylow Subgroups and their factor groups are abelian accounting to the solubility of nth roots of unity by composition series. We combine the classical Euler Formula and the De Moivre Theorem to present the solvability of nth roots of unity. The P-Groups over nth roots of unity are multiplicative. nth roots of unity are subsequences of nth roots of unity and it converges to the limit point of the nth roots of unity.},
year = {2021}
}
TY - JOUR T1 - Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach AU - Alechenu Benard AU - Babayo Muhammed Abdullahi AU - Daniel Eneojo Emmanuel Y1 - 2021/06/29 PY - 2021 N1 - https://doi.org/10.11648/j.eas.20210603.12 DO - 10.11648/j.eas.20210603.12 T2 - Engineering and Applied Sciences JF - Engineering and Applied Sciences JO - Engineering and Applied Sciences SP - 49 EP - 54 PB - Science Publishing Group SN - 2575-1468 UR - https://doi.org/10.11648/j.eas.20210603.12 AB - Solubility of algebraic structures is what gleaned the introduction of group theory, which later stems the other realms of abstract algebra viz: rings, fields and semigroup theories. The nth roots of unity is found in the most sensitive texts ever in the history of abstract algebra: Cauchy’s, Galois’ and Cayley’s. These three giant group theorists had the common ground of the roots of unity in even the title of their works. The idea is that if the nth roots of unity are solvable by radicals and so do the composition series approach, then all other products of the nth roots of unity - which the unity itself is part of - will automatically be solvable. Hence, all equations that dissolve to the least of the nth roots of unity are solvable by the composition series. This article penciled down how the congruence modulo of arithmetics due to Gauss and Leibnitz were used to break down the nth roots of unity, so that the recursive process can generate the composition series of normal subgroups between the unity and the group itself. Since they are P-Groups, they have normal P-Sylow Subgroups. The normality comes from the Index Theorem. Because they all have index 2 in their P-Groups, they are the maximal proper normal P-Sylow Subgroups and their factor groups are abelian accounting to the solubility of nth roots of unity by composition series. We combine the classical Euler Formula and the De Moivre Theorem to present the solvability of nth roots of unity. The P-Groups over nth roots of unity are multiplicative. nth roots of unity are subsequences of nth roots of unity and it converges to the limit point of the nth roots of unity. VL - 6 IS - 3 ER -
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