Show that a polynomial is inside $\Bbb Q^*$

Question

Observe the following polynomial over the polynomial ring $\Bbb Q[x]$: $$f(x)=x^5+1$$ We can rewrite $f$ as $f(x)=(x-1)(x^4-x^3+x^2-x+1)$. Now for showing that $f$ is not reducible, $(x-1) \in \Bbb Q[x]^*$. Now I know that $\Bbb Q[x]^*=\Bbb Q^*$ but I can't use it...

Answer 1

Your factorisation is slightly off; it should be $$f(x)=(x+1)(x^4-x^3+x^2-x+1)$$ The existence of a linear factor already shows by definition that $f$ is reducible.