Odd and even powers cannot cancel

1. A.$\psi$ is surjective.
2. B.If $p_{1}(x)y + p_{2}(x) \in \ker \psi$ for polynomials $p_{1}(x), p_{2}(x) \in \mathbb{C}[x]$, then both $p_{1}$ and $p_{2}$ are the zero polynomial.✓ Correct
3. C.There exist non-zero polynomials $p_{1}(x), p_{2}(x) \in \mathbb{C}[x]$ such that $\psi(p_{1}(x)y + p_{2}(x)) = 0$.
4. D.There exists $f \in \mathbb{C}[x, y]$ such that $\psi(f(x, y)) = 0$ and $\psi(f^{2}(x, y)) \ne 0$.

The trap it tests

The idea was right. The computation was not.