Let be a finite field of order and let be a polynomial in of degree . Show that has order .
In this previous exercise, we showed that every coset in is represented by a polynomial of degree less than , and that two such polynomials represent distinct cosets. Evidently then the number of elements of is precisely the number of polynomials in having degree less than . Each such polynomial is uniquely determined by its coefficients, of which there are , each taking one of the values in . Thus contains elements.