Classify all $A\in M_5(\mathbb{Q}):A^8=I$ and $A^4\not=I$

Question

Classify all $A\in M_5(\mathbb{Q}):A^8=I$ and $A^4\not=I$

Attempt

Since $A^8=I$ it must be $m_A(x)|x^8-1=(x-1)(x+1)(x^2+1)(x^4+1)$. But $A^4\not=I$ so $m_A(x)\not=x-1,x+1,x^2-1,x^2+1,x^4-1,(x-1)(x^2+1),(x+1)(x^2+1)$

And $A\in M_5(\mathbb{Q})\Rightarrow \deg(\chi_A(x))=5$

How should I continue? Hoe would you approach this question?

I want to find the similarity classes using rational normal form.

I know that $\chi_A(x)=$product of the prime divisors of $A$ and $m_A(x)=$lcm of the prime divisors of $A$

Answer 1

Its minimal polynomial, and so its characteristic polynomial $p(x)$, must have $x^4+1$ as a factor. So $p(x)=(x-\alpha)(x^4+1)$ for some $\alpha$. But then $\alpha\in\Bbb Q$ and there aren't many possibilities for it...

In all cases $p(x)$ will be squarefree, so the RCF for $A$ will be the companion matrix of $p(x)$.