Part CCSIR NET June 2025x-fifth-plus-x-plus-1-is-divisible-by-the-cyclotomic-x-squared-plus-x-plus-1

X fifth plus x plus 1 is divisible by the cyclotomic x squared plus x plus 1

Let $f (X) = X^{5} + X + 1 \in Q [X]$ and $g (X) = X^{5} - X + 1 \in Q [X]$ . Which of the following statements are true?

A.f(X) is irreducible in $Q [X]$ , but g(X) is not.
B.g(X) is irreducible in $Q [X]$ , but f(X) is not.✓
C.Both f(X) and g(X) are irreducible in $Q [X]$ .
D.Neither f(X) nor g(X) is irreducible in $Q [X]$ .

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 standard counterexample.

See pricing

50 are analysed free — try those first.

The trap it tests

Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

Drill statements like this

Related counterexample: Irreducible over ℤ ⇒ irreducible mod every prime

X fifth plus x plus 1 is divisible by the cyclotomic x squared plus x plus 1

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