Reducible over any Finite Field but Irreducible over Q

We give a simple argument to show if a = b ∈ Z so that no prime p has the property p2|a or p2|b, then the minimal polynomial of √a+√b over Q, although irreducible over Q by definition, is reducible over any finite field. This provides infinitely many polynomials with the property that they are irreducible over Q yet reducible over any finite field and is a generalization of previous work by M.A. Lee. We also give conditions for the minimal polynomial to have linear factors over Zp and for the minimal polynomial to have multiple roots in an extension field of Zp. Mathematics Subject Classification: 12F10, 02A0