Exercise 6.3.13 (a) Assume p is prime. Show that there are (p - 1)/2 irreducible polynomials of the form f(x) = x – b in Z,[r]. (b) Show that for every prime p, there exists a field with p2 elements....

Extracted text: Exercise 6.3.13 (a) Assume p is prime. Show that there are (p - 1)/2 irreducible polynomials of the form f(x) = x – b in Z,[r]. (b) Show that for every prime p, there exists a field with p2 elements. There is actually a formula for the number of irreducible polynomials of degree d over Z, or any finite field. See Dornhoff and Hohn [25, p. 377].