How to show that $x-1$ and $x^2+x+1$ are irreducible over $\mathbb R[x]?$
I can see that $x – 1$ and $x^2 + x + 1$ are nonzero and non-units (for any field $F$ the set of all units of $F[x]$ = $F-\{0\}$)?
Added: Actually I would like to show $f(x)|(x-1)(x^2+x+1)~\text{where }\deg f(x)\ge 1~\implies f(x)=(x-1)(x^2+x+1)~or~(x-1)~or~(x^2+x+1).$ Does showing $(x-1)~and~(x^2+x+1)$ irreducible help me anyhow?
Best Answer
For the first polynomial, you might want to see what would happen if you could express $x-1=p*q$, where p and q are polynomials with smaller degree. For the second, maybe check out the roots of the quadratic.