Part BCSIR NET December 2024odd-and-even-powers-cannot-cancel

Odd and even powers cannot cancel

Consider the ring homomorphism $ψ : C [x, y] \to C [t]$ defined by $ψ (f (x, y)) = f (t^{2}, t^{3})$ . Which of the following statements is true?

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

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

50 are analysed free — try those first.

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