GATE 2026 MA – Question 23
Over ℤ₁₁, let p(x) = x²+x+4. Which statements are true?
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: (A) p is irreducible over ℤ₁₁.; (B) ℤ₁₁[x]/⟨p⟩ is a field.; (C) ⟨p⟩ is maximal.
Explanation
Discriminant = 1−16 = −15 ≡ 7 modulo 11. Quadratic residues mod 11 are 0,1,3,4,5,9, excluding 7. Thus p has no root and is irreducible. Its ideal in the polynomial ring over a field is maximal; its quotient is a field, and maximal ideals are prime. **A,B,C**.