Theorem (Unique Factorization Theorem for Polynomials)

Every non-constant polynomial $f\in\mathbb{F}[x]$ can be written as a product of a unit and irreducible polynomials. This factorization is unique up to associates. i.e. if $$f=\mathscr{l}_{1}\dots \mathscr{l}_{k}=\mathscr{l}_{1}'\dots \mathscr{l}_{t}'$$ are two different factorizations of $f$ as a product of irreducible polynomials $\mathscr{l}_{i}$ and $\mathscr{l}_{j}'$ with $1\le i\le k$ and $1\le j \le t$, then $k=t$ and (after a permutation) the irreducible factors $\mathscr{l}_{i}$ and $\mathscr{l}_{i}'$ are associates.