On the Reducibility of Cyclotomic Polynomials over Finite Fields

The irreducibility of a polynomial in a prime modulus implies irreducibility over the rationals. However, the converse is certainly not true. In fact, there are some polynomials that are irreducible over the rationals yet reducible in every prime modulus. We show that the nth cyclotomic polynomial reduces modulo all primes if and only if the discriminant of the nth cyclotomic polynomial is a square. We pose further questions about the specific factorization of cyclotomic polynomials over finite fields in relation to the discriminant.