Problem in the "proof" of Eisenstein's criterion on irreducibility.

Suppose $b$ mod $P$ has nonzero constant term. Then $a$ mod $P$ must be identically zero, since otherwise its lowest nonzero coefficient would multiply by the constant term in $b$ mod $P$ to produce a nonzero term of degree less than $n$ in the product $ab$ mod $P$, since $P$ is prime. (It wouldn't be cancelled by anything else since all lower degrees of $a$ mod $P$ are $0$.) But that means $a$ is in $P$, and yet the original polynomial is not in $P$, so this can't happen.