TITLE: Over $\mathbb{F}_2$, find a $5\times 5$ matrix of order 31.
QUESTION [7 upvotes]: So I found that $x^5+x+1$ is irreducible over $\mathbb{F}_2$ and produced the quotient, a field of order $32$. Now, I need to find a $5\times 5$ matrix of order 31 over $\mathbb{F}_2$. I don't see the direction to take here. Could someone point me in the right direction?

REPLY [3 votes]: Take the companion matrix of the irreducible polynomial $x^5+x^2+1$ (thanks @Jyrki Lahtonen):
$$
A = \pmatrix{ 0&0&0&0&1 \\1&0&0&0&0  \\0&1&0&0&1 \\0&0&1&0&0 \\0&0&0&1&0 }.
$$