# Is the field of real numbers a unique factorization domain?

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 $\mathbb{R}$?

• Every field is a UFD. Every non-zero element has the trivial factorization. – pisco Nov 13 '17 at 11:03