Composition Series of the Solvable Multiplicative Abelian Groups over a Regular Even nth Roots of Unity: A Classical Approach

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.