1. In the proof, D&F showed that reducible in if and only if reducible in . Now which recovers the original statement of the theorem.
2. Suppose the gcd of the coefficients of is 1 and for some (so is reducible in ). We want to show that are non-constant.
Suppose otherwise. Then where or is constant (WLOG, take to be constant). Then . Since the gcd of the coefficients of is 1, we necessarily have . Thus which is a contradiction.