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

Compute, up to similarity, all the elements of having order 4. Do the same in .

Our task is to find all the matrices over which satisfy but not or . Note that over , . If is a matrix of order 4, then the minimal polynomial must be divisible by and must divide . Since the characteristic polynomial of has degree 2, there is only one possible list of invariant factors for , namely . The corresponding rational canonical form matrix is . Thus every element of of order 4 is similar to this matrix.

Now over , we have . If is a matrix of order 4, then its minimal polynomial must be divisible by either or , must divide , and must have degree at most 2. There are seven possibilities for the minimal polynomial of , 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.

- ,
- ,

The corresponding rational canonical forms are as follows.

Every matrix in of order 4 is similar (i.e. conjugate) to exactly one matrix in this list.

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