It has a sense to says that a field is an UFD ? (unique factorization domain) For example is $\mathbb Q$ a UFD ? I would say no since for me in a field irreducible element has no sense. Indeed, let $K$ a field. Then $K$ the only ideals are $(1)$ and $(0)$, therefore, the only irreducible are $1$ and $0$.
Am I right ?
Best Answer
Following the wikipedia definition, yes.
Every $x \in K$ with $x \neq 0$ is a unit. There are no irreducible elements. So the "factorisation" of $x$ is just $x$ (a unit, empty product of irreducibles), and this is clearly unique in the sense as mentioned on the wiki page.
They also explicitly mention the inclusion of fields into UFD's.