Lemma: Every element $x\neq 0$ of $\mathcal{O}_K$ with ($K$ an arbitrary number field) can be written as a product of irreducible elements.
Proof: We prove this lemma by complete induction on the norm of $x$. For $N(x) = \pm 1$ we have that $x$ is a unit. Let $|N(x)| \geq 2$ and $x$ not irreducible. Then we can write $x$ as $x = yz$ with $y, z$ not units, that means $|N(y)|, |N(z)|\geq 2$. Thus $|N(x)| > |N(y)|, |N(z)|$ and by the induction hypothesis we can write $y$ and $z$ — and consequently also $x$ — as a product of irreducible elements.
- So this lemma means that I'm guaranteed to come to an end with splitting up $x \in \mathcal{O}_K$ into products of non-units?
- I don't understand the proof. Why shouldn't $y$ and $z$ be irreducible?