# Irreducible Polynomials

Show that there are exactly (p^2-p)/2 monic irreducible polynomials of degree 2 over Z_p, where p is any prime.

Using the definition of irreducibility, Theorem: A polynomial of degree 2 or 3 is irreducible over the field F iff it has no roots in F, or Lemma of Theorem: The nonconstant polynomial p(x) an element of F[x] is irreducible over F iff for all f(x), g(x) an element of F[x], p(x)|(f(x)g(x)) implies p(x)|f(x) or p(x)|g(x), or any other known theorems of polynomials except those involving irreducibility.

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suppose that X^2 + aX + b is irreducible over Z_p. so there are p^2 possibilities for the ...

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This solution is comprised of a detailed explanation to show that there are exactly (p^2-p)/2 monic irreducible polynomials of degree 2 over Z_p, where p is any prime.