Let be a finite dimensional vector space over and suppose is a nonsingular linear transformation on such that . Prove that the dimension of is divisible by 3. If the dimension of is precisely 3, prove that all such transformations are similar.
If is such a transformation, then we have , and so . So the minimal polynomial of divides . Now is irreducible over by the Rational Root Test (Prop. 11 on page 308 in D&F). So the minimal polynomial of is precisely . Now the characteristic polynomial of divides some power of (Prop. 20 in D&F) and so has degree for some . But of course the degree of the characteristic polynomial is precisely the dimension of , as desired.
Now if has dimension 3, then the minimal polynomial of is the characteristic polynomial, so that the invariant factors of are simply . In particular, all such are similar.