# Thread: Probably a simple explanation

1. ## Probably a simple explanation

Could someone explain why $2x-10$ is not an irreducible in $Z[x]$ but is an irreducible in $Q[x]$.

2. Originally Posted by Ryaη
Could someone explain why $2x-10$ is not an irreducible in $Z[x]$ but is an irreducible in $Q[x]$.
It factorises as 2(x–5) in both cases. However, 2 is a unit in $\mathbb{Q}[x]$ but not in $\mathbb{Z}[x]$.

3. Originally Posted by Opalg
It factorises as 2(x–5) in both cases. However, 2 is a unit in $\mathbb{Q}[x]$ but not in $\mathbb{Z}[x]$.
Why would that not be the other way around? (since it is irreducible in $Q[x]$ and reducible in $Z[x]$)

4. Originally Posted by Ryaη
Why would that not be the other way around? (since it is irreducible in $Q[x]$ and reducible in $Z[x]$)
Reducible means having a nontrivial factorisation. Irreducible means having no nontrivial factorisation. Nontrivial means that neither of the factors is a unit. If one of the factors is a unit then the factorisation doesn't count as a nontrivial factorisation. Since 2 is a unit in $\mathbb{Q}$, the factorisation 2x – 10 = 2(x – 5) doesn't count as nontrivial. That is why 2x – 10 is irreducible in $\mathbb{Q}[x]$.