Let be a prime number. We know that . Let be the ring the polynomials with indeterminate and coefficients in . The polynomial factors over as follows for various :
None of the factors above can be factored anymore, hence they are irreducible over . Let us call the trivial factor since the root already belongs to . But why do we have two nontrivial irreducible factors of , each of degree 3, whereas has only one non-trivial irreducible factor of degree 12? It appears that either there is only one nontrivial factor, or the degree of all nontrivial factors is the same. Why?