Special Project: Factoring over $\mathbb{Q}$ or $\mathbb{Z}$

Factoring a polynomial in ${\mathbb{Q}}[x]$ is essentially equivalent to polynomial in $\mathbb{Z}[x]$, since one can always multiply by a suitable large integer to clear all the denominators. However, the situation is even better than this. Remarkably enough, it is not possible to factor a polynomial with integer coefficients into polynomials with rational (and non-integral) coefficients. This fact is proven in the next subsection.