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

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- Mar 31st 2010, 09:45 AMRyaηProbably a simple explanation
Could someone explain why $\displaystyle 2x-10$ is not an irreducible in $\displaystyle Z[x]$ but is an irreducible in $\displaystyle Q[x]$.

- Mar 31st 2010, 12:02 PMOpalg
- Mar 31st 2010, 08:55 PMRyaη
- Apr 1st 2010, 12:11 AMOpalg
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 $\displaystyle \mathbb{Q}$, the factorisation 2x – 10 = 2(x – 5) doesn't count as nontrivial. That is why 2x – 10 is irreducible in $\displaystyle \mathbb{Q}[x]$.