## Exhibit the matrices of dimension 2 over QQ having multiplicative order 4

Compute, up to similarity, all the elements of $\mathsf{GL}_2(\mathbb{Q})$ having order 4. Do the same in $\mathsf{GL}_2(\mathbb{C})$.

Our task is to find all the $2 \times 2$ matrices over $\mathbb{Q}$ which satisfy $p(x) = x^4-1$ but not $x^2-1$ or $x-1$. Note that over $\mathbb{Q}$, $p(x) = (x^2+1)(x+1)(x-1)$. If $A$ is a matrix of order 4, then the minimal polynomial must be divisible by $x^2+1$ and must divide $x^4-1$. Since the characteristic polynomial of $A$ has degree 2, there is only one possible list of invariant factors for $A$, namely $x^2+1$. The corresponding rational canonical form matrix is $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. Thus every element of $\mathsf{GL}_2(\mathbb{Q})$ of order 4 is similar to this matrix.

Now over $\mathbb{C}$, we have $p(x) = (x+1)(x-1)(x+i)(x-i)$. If $A$ is a matrix of order 4, then its minimal polynomial must be divisible by either $x+i$ or $x-i$, must divide $p(x)$, and must have degree at most 2. There are seven possibilities for the minimal polynomial of $A$, and with the minimal polynomial chosen, the remaining invariant factors are determined (since the characteristic polynomial has degree 2). The possible lists of invariant factors are as follows.

1. $x+i$, $x+i$
2. $(x+i)(x-i)$
3. $(x+i)(x+1)$
4. $(x+i)(x-1)$
5. $x-i$, $x-i$
6. $(x-i)(x+1)$
7. $(x-i)(x-1)$

The corresponding rational canonical forms are as follows.

1. $\begin{bmatrix} -i & 0 \\ 0 & -i \end{bmatrix}$
2. $\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$
3. $\begin{bmatrix} 0 & -i \\ 1 & -1-i \end{bmatrix}$
4. $\begin{bmatrix} 0 & i \\ 1 & 1-i \end{bmatrix}$
5. $\begin{bmatrix} i & 0 \\ 0 & i \end{bmatrix}$
6. $\begin{bmatrix} 0 & i \\ 1 & i-1 \end{bmatrix}$
7. $\begin{bmatrix} 0 & -i \\ 1 & 1+i \end{bmatrix}$

Every matrix in $\mathsf{GL}_2(\mathbb{C})$ of order 4 is similar (i.e. conjugate) to exactly one matrix in this list.