Let p be a prime, n an integer distinct from 0, and d an irreducible polynomial in Z/pZ[X] of degree n. We are concerned with the finite field S = Z/pZ[X]/(d).


Theorem

Write q = pn for the cardinality of S. Then, for each a, b S,

  1. a + a + ··· + a   (p terms) = 0.
  2. (a + b)p = ap + bp.
  3. aq = a.

The theorem implies that every nonzero element in S raised to the power q - 1 is equal to 1.

An element of S having no smaller (positive) power equal to 1 is called primitive. In general, the smallest positive number l satisfying al = 1 is called the order of a. So a nonzero element of S is primitive if its order is q - 1.


Fact

S has a primitive element.


This result directly generalizes the corresponding fact for Z/(p).