The $5\times 5$ matrix $A$ is a [circulant matrix](https://en.wikipedia.org/wiki/Circulant_matrix) and so is $A-I$ and thus $\, p(x,y) := \det(A-I)\,$ factors into a product of $5$ linear factors each of the form $\, ax^5 + by^5 \,$
where $\, a,b \in \mathbb C. \,$ Each of these factors into a product of $5$ linear factors of the form $\, cx + dy \,$ where $\, c,d \in \mathbb C.\,$ 

This is not correct. Maybe I can fix it....