# Why is $1$ not an irreducible integer?

When defining irreducible integers, we restrict our attention $Z = \{n \in \mathbb{Z} \; | \; |n| > 1\}$. Then we say that for some $p \in Z$, $p$ is irreducible if for some $a,b \in \mathbb{Z}$: $$p = ab \implies |a| = 1 \;\vee\; |b| = 1$$

Why do we exclude $1$ and $-1$ from the definition? Wouldn't they fit the condition?

If 1 were a prime, the number of possible prime factors in a given integer is unbounded, as $1^n$ divides anything.