<s>[INST] Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^5+x+1$ Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^5+x+1$
Attempt:
We can write $x^5+x+1 = (x^2+x+1)(x^3-x^2+1)$ which could be a hint to be reducible, if we look clously we see that $(x^2+x+1),(x^3-x^2+1)$ are both irreducible in $Q$, if we use rational root theorem, we see there is no root in $Q$, so its irreducible over $Q$?
 [/INST] Let $x=w,w^2$, where $w$ cube root of unity. The $f(x)=x^5+x+1 \implies f(w)=w^2+w+1=0, f(w^2)=w+w^2+1=0$. So $g(x)=(x-w)(x-w^2)=x^2+x+1$ is a factor of $f(x)$ Further $\frac{f(x)}{g(x)}=x^3-x^2+1.$
So $f(x)$ is reducible.
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