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?
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