## Unique Factorization Domains

Abbreviation: UFDom

### Definition

A unique factorization domain is an integral domains $D$ such that

every element is a product of irreducibles: $\forall a\in D \exists p_1,...,p_r\in D, n_1,...,n_r\in \mathbb{N}$ such that $a=p_1^{n_1}\cdotp_2^{n_2}...p_r^{n_r}$ and $p_i$ is irreducible for $i=1,\ldots,r$

the product is unique up to associates: $\forall \mbox{ irreducibles } p_i,q_j$ if $a=p_1^{n_1}\cdot p_2^{n_2}...p_r^{n_r}=q_1^{m_1}\cdot q_2^{m_2}...q_s^{m_s}$ then $r=s$ and each $p_i$ is an associate of some $q_j$

### Examples

Example 1: $\mathbb{Z}[x]$, the ring of polynomials with integer coefficients.

### Properties

Classtype second-order

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &1\\ f(5)= &1\\ f(6)= &0\\ \end{array}$