Is the following statement true? In every integral domain, irreducible implies prime

No. 2 in ℤ[√−5] 2 is irreducible (no element of norm 2) but divides (1+√−5)(1−√−5) = 6 without dividing either factor.

Counterexample

Is this true?

In every integral domain, irreducible implies prime

No — it is false.

The counterexample

2 in $\mathbb{Z}[\surd-5]$

Why it fails

2 is irreducible (no element of norm 2) but divides $(1+\surd-5)(1-\surd-5) = 6$ without dividing either factor.

Keep practicing

Finite-dimensional intuition

Something true in ℝⁿ, or in a nice space, assumed in general.

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