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I believe I might be having a bit of trouble with a question that asks me to find the values of $a$ and $b$ such that the polynomial $x^2-bx-a$ is irreducible and the matrix $\begin{bmatrix} 0 & a \\ 1 & b \end{bmatrix}$ has order $8$.

This is not a homework question in any class that I am taking. This is regarding a research project that I am planning on working with a professor for which I have almost no clue what I am doing.

Did
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asdf
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    Not sure your justifications for including zero personal input hold water but... The polynomial $x^2-bx-a$ is the characteristic polynomial of the matrix hence the hypothesis that the eighth power of this matrix is the identity and that $x^2-bx-a$ is irreducible, implies that $x^2-bx-a$ divides $x^8-1$. Can you finish? – Did Sep 23 '18 at 18:48
  • Working on it. Will get back to you as soon as I am done. Thanks! – asdf Sep 23 '18 at 22:43

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