This question sucks.

1) Isolate the highest power of the polynomial modulo
2) If possible, apply the modulo to turn any negative terms into their equivalent positive term
3) Let $x = \alpha$
4) Express the powers of alpha in terms of the polynomial modulo 5) If you get a remainder of $1$ for a power that is not $n^p - 1$, try a different alpha substitution ($\alpha+1$ ?!?!)

Exponential Arithmetic

  • $x^a * x^b = x^{a+b}$
  • $(x^a)^b = x^ab$
  • $x^{-a} = x^{highestpower - a}

Can express large powers as combinations of smaller powers

  • A field has dimension $n$
  • A field has $n^p$ vectors

Example