05 - Ring of Polynomials with Integer Coefficients is a UFD

Ring of Polynomials with Integer Coefficients is a UFD

Let $R=\mathbb{Z}[t]$ then

$R$ is a UFD

Proof

Since $\mathbb{Z}[t]$ is a integral domain as $\mathbb{Z}$ is also an integral domain
Let $f\in \mathbb{Z}[t]$ then

$$f=a.f^{\prime }\text{ where }c(f^{\prime })=1$$

Since $\mathbb{Z}$ is a UFD then
$a$ is factorisable into a product of prime elements of $\mathbb{Z}$ (prime in $\mathbb{Z}[t]$)
Hence assume $c(f)=1$

As $f$ is an element of $\mathbb{Q}[t]$ then it is a product of prime elements in $\mathbb{Q}[t]$ then

$$f=p_1{p}_2\cdots p_k$$

By Uniqueness of content of polynomial in $\mathbb{Q}[t]$ then

$$p_i=a_i{q}_i\text{ where }{a}_i\in \mathbb{Q}\text{ and }{q}_i\in \mathbb{Z}[t]\text{ and }c(q_i)=1$$

By Property of prime elements (3) as $q_i$ is prime in $\mathbb{Z}[t]$ and

$$f=(a_1\cdots a_k)(q_1\cdots q_k)$$

By comparing contents then

$$(a_1\cdots a_k)=1$$

So then by Equivalent statements for an unique factorisation domain so $R$ is UFD