Similarity classes for matrices

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Homework Statement


How would we find number of similarity classes for a nxn matrix over the field Fp (cyclic of order p) for n=2,3,4?


Homework Equations


A and B are similar iff they have the same monic invariant factors


The Attempt at a Solution


I would say 4 classes for n=2, since all that is really important is if the characteristic polynomial factors over Fp or not. Should the answer depend on p? Or is this just taking cases?
Thanks for any help!
 

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