Does $A$ a UFD imply that $A[T]$ is also a UFD?

I'm trying to prove that $A$ a UFD implies that $A[T]$ is a UFD.

The only thing I am sure I could try to use is Gauss's lemma.

Also, how can we deduce that the polynomial rings $\mathbb{Z}[x_1,\ldots,x_n]$ and $k[x_1,\ldots,x_n]$ are UFDs?

Solution 1:

You should know that if $k$ is a field, then $k[x]$ is a UFD (even a Euclidean domain). So given $g(x)\in A[x]$, first write it as $g(x) = cG(x)$, where $c$ is a constant and $G(x)$ is primitive. Then show that a primitive polynomial in $A[x]$ is irreducible if and only if it is irreducible when viewed as a polynomial in $k[x]$, where $k$ is the field of fractions of $A$. Then use this to take an arbitrary polynomial in $A[x]$, and factor it by "factoring out the content, then factoring it over $k[x]$, and then "lifting the factorization" back to $A[x]$. The argument is exactly the same as that used in the case of $\mathbb{Z}[x]$ with Gauss's Lemma.

Now, since $\mathbb{Z}$ is a UFD, then by this argument so is $\mathbb{Z}[x_1]$; which means that so is $\mathbb{Z}[x_1][x_2]\cong\mathbb{Z}[x_1,x_2]$. Which means that so is $\mathbb{Z}[x_1,x_2][x_3] = \mathbb{Z}[x_1,x_2,x_3]$. Etc. The same holds for polynomials over a field, since a field is trivially a UFD (or because you know that the first polynomial ring, $k[x_1]$, is a UFD).

Solution 2:

There is a slick general way to do this by localization (usually credited to Nagata). Suppose $\rm\:D\:$ is an atomic domain, i.e. nonzero nonunits factor into atoms (irreducibles). If $\rm\:S\:$ is a saturated submonoid of $\rm\:D^*$ (i.e. $\rm\,cd\in S\!\iff c,d\in S)\,$ and, furthermore, $\rm\,S\,$ is generated by primes, then $\rm\: D_S$ UFD $\rm\:\Rightarrow\:D$ UFD. $ $ This is often called Nagata's Lemma.

This yields said slick proof of $\rm\:D$ UFD $\,\rm\Rightarrow D[x]$ UFD, viz. $\rm\:S = D^*\:$ is generated by primes, so localizing yields the UFD $\rm\:F[x],\:$ $\rm F =\:$ fraction field of $\rm\:D.\:$ Hence $\rm\:D[x]\:$ is a UFD, by Nagata.

This yields a more conceptual / structural view of the essence of the matter (vs. the traditional argument by Gauss' Lemma). Moreover, the proof generalizes to various rings closely related to GCD domains, e.g. Riesz/Schreier rings, which provide refinement-based views of UFDs (which prove more convenient for noncommutative generalizations).