Determine (up to similarity) the elements of order 5 in .
It suffices to find the possible rational canonical forms of matrices over whose minimal polynomials divide .
To accomplish this, we must first factor over . To this end, note that .
It is straightforward (if tedious) to show that has no roots in , and so has no linear factors.
Suppose has a quadratic factor; say . Comparing coefficients, we have (i) , (2) , (3) , and (4) . Without loss of generality, we can assume that . (that is, the canonical representatives of and in .) There are then ten possibilities for : , , , , , , , , , and . Substituting (i) into (ii) and (iii) we have (iv) and (v) . If , we may solve (v) for : . Substituting into (iv) and simplifying, we have (vi) . It is straightforward to show that none of our candidate pairs with satisfy this equation. (I suggest using a calculator to verify this; for instance, WolframAlpha can do it with this input. So if does factor into do quadratics, then is either or . Suppose ; then from our coefficient equations (i) – (iv), we have and , and so . Indeed, we can see that over , , so that either or . Now is (by symmetry), and indeed we can verify that .
It is straightforward (if tedious) to show that these quadratic factors are irrducible over since they do not have roots. So completely factors over as .
If is a matrix over of order 5, then its minimal polynomial has degree at most 2, divides , and does not divide . There are two such divisors, which each comprise the entire list of invariant factors of . There are thus two corresponding rational canonical form matrices: and . Every element of order 5 in is similar to exactly one of these two.