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 $\mathbb{Z}[X]$ . Which of the following statements are true?

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

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, and why this one tests base field or ring.

See pricing

50 are analysed free — try those first.

The trap it tests

Base field or ring

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

Drill statements like this

Related counterexample: Irreducible over ℤ ⇒ irreducible mod every prime

Irreducible over Q not mod p

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