Wrong(?) Question about Irreducibiliity of Polynomials
Here is a question from my Galois Theory course:
Let f be a polynomial over with degree 2 or 3, and suppose it is reducible over . What is wrong with the following argument: f is reducible over by Gauss's Lemma and so must have a zero over [Give both a counterexample and show where the error in the proof is].
But wait a minute! Isn't the statement above true by the contrapositive of Gauss's Lemma (i.e. f irreducible over Z => f irreducible over Q)? And isn't it true anyway since the ring of integers of Q is Z, and hence by definition since f has coefficients in Z and is factorisable, its roots will lie in the algebraic integers, which are the integers ? Thanks if anyone could help me with this! :)