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Every field $\mathbb{F}$, with the norm function $\phi(x) = 1, \forall x \in \mathbb{F}$ is a Euclidean domain. Every Euclidean domain is a unique factorization domain.
So, it means that $\mathbb{R}$ is a UFD?
What are the irreducible elements of $\mathbb{R}$?
The (most common) definition of UFD requires that every non-zero non-unit element has a unique factorisation into irreducible elements. A field has no non-zero non-unit elements, so that condition is vacuously satisfied.
No irreducible (non-unit) elements are needed, because we have no non-unit elements that need factorisations.
