# irreducible polynomials

If $f(x)$ is irreducible the proof is complete. Otherwise $f(x) = p_1(x)p_2(x)$ where $p_1(x),p_2(x)$ are not constant. If these two polynomials are irreducible the proof is complete. Otherwise keep on going. Eventually you end up with only irreducible polynomials. Because the sums of the degrees need to add up to $\deg f(x)$ and this means you cannot do this indefinitely.