GATE 2026 MA – Question 52
$T:\mathbb Q[x]\to\mathbb Q[x]$ sends p(x) to p(x²). Which statements hold?
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Correct answer: (A) T is a ring homomorphism.; (B) T is injective.
Explanation
Substitution preserves addition, multiplication and the identity, so A holds. Distinct polynomials remain distinct because powers xᵏ become distinct powers x²ᵏ, proving B. Every image polynomial has only even powers, so x has no preimage. Thus T is not onto or an isomorphism. **A,B**.