The field with nine elements

Let F9 denote the field of nine elements.

  1. Show that each nonzero aF9 is a root of X31=(X1)(X2+1)(X4+1)F3[X].
  2. Use the Pigeonhole Principle to prove that F9 has an element of multiplicative order 8. (Include a proof that the Pigeonhole Principle applies.)