Q:

Eisenstein criterion on f(x+1)

I need to show that the polynomial $f(x)=x^6+x^5+x^4+x^3+x^2+x+1$ is irreducible in $\mathbb{Z}[X]$ and in $\mathbb{F}_2[X]$. As we can't find a prime number p satisfying the conditions for the Eisenstein criterion, I did not know how to solve it. I looked into the solutions and they apply the Eisenstein criterion to $f(x+1)$ instead of $f(x)$. I don't understand why we can do this.
Could somebody explain this to me? And is proving irreducibility for $f(x+1)$ enough?

A:

I looked into the solutions and they apply the Eisenstein criterion to $f(x+1)$ instead of $f(x)$. I don't understand why we can do this.

It suffices to show irreducibility of $f(x+1)$. To see this, assume that $f(x)$ is reducible, then $f(x) = g(x)h(x)$ for some proper factors $g(x)$ and $h(x)$. In that case, you get that
$$f(x+1) = g(x+1)h(x+1)$$
where the right factors are still proper factors. Thus, reducibility of $f(x)$ implies that of $f(x+1)$. The contrapositive shows that it's sufficient to prove the irreducibility of $f(x)$.

