Why $a^3 + b^3 \neq c^3$ for $a, b, c \in \mathbb{Z}$ in one page

This is a ground-up proof by contradiction (requiring no more than high school knowledge to understand) of the famous Fermat's Last Theorem for a special case of n=3 but done in using the special Euclidean Ring $\mathbb{Z}[p]$.