Is the following statement true? A polynomial irreducible over ℚ is irreducible modulo some prime

No. x⁴ + 1 Irreducible over ℚ, yet reducible modulo every prime.

Counterexample

Is this true?

A polynomial irreducible over $\mathbb{Q}$ is irreducible modulo some prime

No — it is false.

The counterexample

$x^{4} + 1$

Why it fails

Irreducible over $\mathbb{Q}$ , yet reducible modulo every prime.

Keep practicing

Standard counterexample

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

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.