is $f(x)=x^4 +x^3 +x^2 +x +1$ irreducible in $R=(\mathbb Z/7\mathbb Z)[x]$? is $f(x)=x^4 +x^3 +x^2 +x +1$ irreducible in $R=(\mathbb Z/7\mathbb Z)[x]$?
I know that $f(x)$ cannot be expressed as a product of degree 1 polynomial and degree 3 polynomial since it has no roots in $R$ but what about two degree 2 polynimials?

Since $(X-1)(X^4 + X^3 + X^2 + X + 1) = X^5 - 1$, if $f$ wasn’t irreducible, you’d have a degree two factor $(X - α)(X - β) = X^2 - (α+β) X + αβ~$ of $X^5 - 1$ of $f$ in $ℤ/7ℤ[X]$, where $α$, $β$ and therefore $αβ$ ($∈ ℤ/7ℤ$) had to be $5$-th roots of unity. What $5$-th roots of unity are there in $ℤ/7ℤ$? What possibilities are there for $α$ and $β$? (Note that $α$ and $β$ don’t have to be elements of $ℤ/7ℤ$ a priori, but elements of some extension of $ℤ/7ℤ$.)
