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.

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