On the Cubic Roots of Unity, and the Additive Structure of the Group of Units Mod

The additive structure of multiplicative semigroup Z(.) mod in residue ring = (+, .) mod (odd prime ) is analysed. The group of units in (.) is cyclic of order ( − 1), with inner product = of || = − 1 coprime to |} = . The -th power units form a subgroup of order ||/, with ||/|| = ( ≥ 2), so = only for =2. Some additive properties of these subgroups of units are derived.The normed equivalence + = −1 mod in subgroup has for =1 mod 6 the cubic root solution + 1 = − = −. This is generalized for any odd prime and > 1 by considering the two arithmetic symmetries − and as commuting functions: () = − and () = 1/ ( = ). Combined with successor function () = + 1 they yield four distinct compositions each of which has period 3 if iterated, e.g. () = −( + 1) and ()()() = for all ≠ 0, −1. This ‘fixed-point' property implies a structure in of three inverse pairs , indices mod 3, where ≡ 1 mod . Restricted to subgroup , this is shown to be the general normal form solution of + ≡ mod