Proving $L$ is the splitting field of $f(x) = \min_Q(ζ_7)$

Question

Given the extension $L = Q(ζ_7)$ of $Q$ where $ζ_7 = e^{2πi/7}$, I had a small difficulty in first proving $L$ is the splitting field of $f(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1$ over $Q$ and that $f(x) = min_Q(ζ_7)$ before showing a further result here Show $ Gal(L|Q)\cong (Z/7Z)^*$ and determine $L_H $

So, for this, I know the six roots of $f(x)$ are $\zeta_{7},\zeta_{7}^{2},\zeta_{7}^{3},\zeta_{7}^{4},\zeta_{7}^{5}$ and $\zeta_{7}^{6}$ so $L = Q(ζ_7)$ is the splitting field of $f(x)$ over $Q$ but how to show $f(x)$ is irreducible over $Q$? I must be missing something simple here.

Answer 1

If $f(x)$ was reducible, $f(x+1)$ would be reducible too. But you can apply Eisenstein's criterion to it, with $p=7$, since$$f(x+1)=x^6+7x^5+21x^4+35x^3+35x^2+21x+7.$$Of course, the same argument proves that, for any prime $p$, the polynomial$$x^{p-1}+x^{p-2}+\cdots+x+1$$is irreducible in $\mathbb{Q}[x]$.