Investigate if the following polynomial in Q [x] is reducible or irreducible: $x^5+x+1$

Question

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 $Q$?

Answer 1

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.

Answer 2

Since it can be factored into two polynomials in $\mathbb Q[x]$, $x^5+x+1$ is reducible. That the two factors have no rational root is irrelevant.