Is this $x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ irreducible in $\mathbb Q[x]$?

Question

Is this $x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ irreducible in $\mathbb Q[x]$? Can you help please?

Answer 1

Let $f(x)=x^6+x^5+x^4+x^3+x^2+x+1$. Define the new polynomial $f(x+1)=x^6+7x^5+21x^4+35x^2+21x+7$, using the binomial theorem expand $f(x+1)$. This polynomial is irreducible over $\mathbb{Q}$ by Eisenstein criterion, for $p=7$

If $f(x)$ were reducible over the rationals then let $p(x), q(x)\in \mathbb{Q}[x]$ such that $$f(x)=p(x)q(x)$$ where $\deg(p)< 6$ and $\deg(q)<6$ We would then have $$f(x+1)=p(x+1)q(x+1)$$ but that's impossible! Hence $x^6+x^5+x^4+x^3+x^2+x+1$ is irreducible over $\mathbb{Q}[x]$