is f(x) = 0 irreducible in $\mathbb{Z} /2 \mathbb{Z}$?

Question

Let's say I have a polynomial like $$f(x) = 4x^2 +12x +28$$ when I reduce this with respect to mod 2; I end up with $0$.

Can I say that zero is irreducible in $\mathbb{Z} /2 \mathbb{Z}$ so $f(x)$ is irreducible in $\mathbb{Q}$ ?

Answer 1

An irreducible polynomial is a polynomial of degree $\ge1$ that cannot be factored into two polynomials of degree $\ge 1$. Under this definition, the $0$ polynomial is not irreducible.