The $5\times 5$ matrix $A$ is a 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....