I want to prove that the following polynomial is irreducible: $$x^3 – x^2 – x + 3$$
My question gives the hint to apply the substitution $x \mapsto x+1$ but I've tried this and when multiplied out I'm getting $x^3 + x^2 +2$. I tried this mod 2 but came out with a reducible polynomial ($x^3 + x^2$ which can be written $(x^2)(x+1)$).
Can anyone tell me what I'm missing? I know that irreducible mod prime number implies irreducible in $\mathbb Z$, but am I wrong in thinking it works the other way i.e. reducible mod prime means reducible in $\mathbb Z$?
Best Answer
The thing is that with $f(x)=x^3-x^2-x+3$ you get $$ f(x+1)=x^3+2x^2+2 $$ and we are good...