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    D|A D B C

It's easy to see that this is indeed a *good* multiplicative table for a field.

## Irreducible Polynomial of Order \\(n\\)?

Now that we have a trick, we still have to find out one last way to
actually find an irreducible polynomial of the desired order \\(n\\) over
the field we are extending. There are some results about it:

- they actually exist! There is someone that calculated a formula to count
  them, which also implies that fields of order \\(p^n\\) exists, of
  course! See [this question][irred-count] for further information;
- there always exist *monic* ones, i.e. where the coefficient for the
  highest power of \\(x\\) is \\(1\\) (which simplifies the division and
  the rest calculation);
- there's more than one way to test for the irreducibility of
  a polynomial... but we only need one, of course.

For the second bullet, we will refer to [Rabin's test for
irreducibility][rabin-test] with the algorithm that follows (in Perl)