Is $1+x+x^2+x^3+x^4$ irreducible over $\mathbb{Z}$?

Question

Is $1+x+x^2+x^3+x^4$ irreducible over $\mathbb{Z}$?

I understand that if asked for $\mathbb{Q}$, the answer is yes, because $$f(x) = x^{5-1}+ x^{5-2} + x^{5-3} +x^{5-4} +1.$$

Since $5$ is prime, therefore it is irreducible over $\mathbb{Q}$.

But can we say that, since it is irreducible over $\mathbb{Q}$, it is also irreducible over $\mathbb{Z}$? I am not clear about it.

Answer 1

$$f(x) = 1+x+x^2+x^3+x^4$$

$$f(x) = x^{5-1}+ x^{5-2} + x^{5-3} +x^{5-4} +1.$$

Since $5$ is prime, therefore it is irreducible over $\mathbb{Q}$.

By Gauss's lemma,

A primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers.

Using the lemma above,

Since $1+x+x^2+x^3+x^4$ irreducible over $\mathbb{Q}$, therefore, $1+x+x^2+x^3+x^4$ irreducible over $\mathbb{Z}$ too.

Answer 2

Yes, of course.

Another way:

Let $x=y+1$.

Thus, $$x^4+x^3+x^2+x+1=y^4+5y^3+10y^2+10y+5$$ and use Eisenstein: https://en.wikipedia.org/wiki/Eisenstein%27s_criterion