Unique factorization theorem

The polynomial analog of the fundamental theorem of arithmetic for the integers is the unique factorization theorem.

Theorem 2.4.4 (Unique factorization theorem)   Let $F$ be a field and $p(x)\in F[x]$. Then there are monic irreducible polynomials $p_i(x)\in F[x]$, integers $e_i>0$, $i=1,2,...,k$, and a constant $c\in F^\times$ such that

$$f(x)=cp_1(x)^{e_1}...p_k(x)^{e_k}.$$

Furthermore, this expression is unique up to order.

proof: The proof is analogous to the proof of the fundamental theorem of arithmetic. It is therefore omitted. $\Box$