Difference of Two Powers

Theorem
Let $\mathbb F$ denote one of the standard number systems, that is $\Z$, $\Q$, $\R$ and $\C$.

Let $n \in \N$ such that $n \ge 2$.

Then for all $a, b \in \mathbb F$:

For convenience of applicability, these results are sometimes separated into two cases for odd and even indices:

General Commutative Ring
The result can also be extended into the general abstract algebraic context as follows: