The usual proof for unique factorization in $\mathbb{Z}$ proceeds via the concept of GCD (greatest common divisor) of two integers leading to the fundamental property of primes in $\mathbb{Z}$:
Theorem: If $p$ is prime and $a, b \in \mathbb{Z}$ such that $p\mid ab$ then either $p\mid a$ or $p\mid b$.
And this is the result which guarantees uniqueness of factorization. Note that the same procedure does not apply in $\mathbb{Z}[x]$ as we don't have a concept of GCD here. Thus for example we can't say that $2, x$ have GCD $1$ because we can't find polynomials $p(x), q(x) \in \mathbb{Z}[x]$ such that $$1 = 2p(x) + xq(x)$$ (contradiction arises when we put $x = 0$ in above equation).
The problem is eliminated by considering polynomials in $\mathbb{Q}[x]$ and then we have the GCD available here so that $\mathbb{Q}[x]$ is a unique factorization domain. Next we use the fact that $\mathbb{Z}[x]\subset\mathbb{Q}[x]$ to factorize elements of $\mathbb{Z}[x]$ as product of polynomials in $\mathbb{Q}[x]$ and then use Gauss lemma to prove that the factorization can also be done using polynomials in $\mathbb{Z}[x]$ only.
Thus it appears that existence of GCD is not necessary to guarantee existence of unique factorization. Does that mean we can prove unique factorization in $\mathbb{Z}$ via some other means rather than the approach I outlined in the beginning?
Also I would like to know if the property of prime numbers (mentioned in theorem at the beginning) is always a consequence of the existence of GCD in a more generally setting of integral domains?
Update: From the comments and in particular the wiki link given by Hand Lundmark it is clear that the ideas of the usual ring of integers have been generalized in many ways to give rise to the famous chain of class inclusions (see wiki link on GCD Domains) and this question is perhaps a very naive attempt to understand that all (or some of) those inclusions are proper.
Further Update: I was a bit hesitant about asking question related to a seemingly trivial matter (namely unique factorization in integers and polynomials with integer coefficients), but the way it has been received here is so much more than what I expected. MSE never ceases to amaze me (and perhaps other users too)! Thanks to all those who answered/commented/chatted. I have now got a lot of food for thought (and study).
Best Answer
Effectively there is not necessary the existence of GCD in order to prove the fundamental theorem of arithmetic. It's enough a weaker condition, namely we only need the four number lemma, see e.g., this answer. This lemma in a more abstract setting give rise to two special classes of domains known as Schreier domains and pre-Schreier domains (or Riesz domains).
Egreg wrote:
Domains satisfying the first condition are called ACCP domains, whereas the domains satisfying the second condition are called AP-domains (atom $\implies$ prime), where atom=irreducible.
An important result is the following:
Theorem: If $R$ is an ACCP domain, then $R[x]$ is also an ACPP domain.
Proof: This is theorem 17a) in Pete L. Clark's notes on factorization in integral domains.
I'm not aware of any characterization of AP domains, but what I know is that every GCD domain is an AP-domain, even more: both Schreier and pre-Schreier domains are AP-domains and more exactly we have the following chain:
$$\text{GCD-domain} \implies \text{Schreier domain} \implies \text{pre-Schreier domain} \implies \text{AP-domain}.$$
Therefore we have the following characterization of unique factorization domains:
$$\text{UFD} \iff \text{ACCP domain}+\text{Schreier domain}\; (^*)$$
On the other hand, it turn out the being a Schreier domain behaves well under polynomial ring extensions, namely we have the following:
Theorem (Cohn): If $R$ is a Schreier domain, then so is $R[x]$.
Proof: This is theorem 2.7 in the paper Bezout rings and their subrings.
So we can apply (*) to the particular cases of $\Bbb Z$ and $\Bbb Z[x]$ in this way:
i) For $\Bbb Z$:
ii) For $\Bbb Z[x]$:
Since $\Bbb Z$ satisfies the ACCP condition, then $\Bbb Z[x]$ also satisfies the ACCP condition, so this will give us the existence of the irreducible factorization.
Since $\Bbb Z$ is a Schreier domain, then $\Bbb Z[x]$ is also a Schrier domain, so this will guarantee the uniqueness.
However, there is a little more to say. The classical proof of: $$D\; \text{UFD} \implies D[x]\; \text{UFD}\; (^{**})$$ uses, as you've noted, Gauss' lemma: the product of two primitive polynomials is primitive, but the above result isn't necessary because we can proof (**) without it. For instance, you can check theorem 27 of Pete L. Clark's notes cited lines above.
Returning to Gauss' lemma, this result gives rise to the class of domains known as GL-domains, and Anderson and Zafrullah proved that they are between the class of pre-Schreier and AP-domains. So in conclusion we have the chain:
$$\text{GCD-domain} \implies \text{Schreier domain} \implies \text{pre-Schreier domain}$$ $$\implies \text{GL-domain} \implies \text{AP-domain}.$$
For more info related to Schreier, pre-Schreier and GL-domains you can check these papers: