Part CCSIR NET December 2023irreducible-over-Q-not-mod-p

Irreducible over Q not mod p

Let $f (X) = X^{2} + X + 1$ and $g (X) = X^{2} + X - 2$ be polynomials in $Z [X]$ . Which of the following statements are true?

A.For all prime numbers p, f(X) mod p is irreducible in $(Z / p Z) [X]$ .
B.There exists a prime number p such that g(X) mod p is irreducible in $(Z / p Z) [X]$ .
C.g(X) is irreducible in $Q [X]$ .
D.f(X) is irreducible in $Q [X]$ .

The answer and working come with the PYQ pack.

The trap it tests

Base field or ring

The answer changes with the field or ring you are working over.

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