Unique factorization in the

polynomial ring in

*n* variables
is a crucial result for

number theory and

algebraic geometry. (The proof relies heavily
on

Gauss's Lemma so you might want to read that first.)

**Theorem** If *R* is a unique factorization domain
then so is *R[x]*.

**Proof:**
Let *k* be the field of fractions of *R*.
We know that *k[x]* is a UFD since it is a Euclidean ring.
Now let *f* in *R[x]* be nonzero. Since
*k[x]* is a UFD we can write

*f=cf*_{1}...f_{r}

for *c* in *k* and each *f*_{i} irreducible
in *k[x]*. By adjusting *c* we can arrange
that each *f*_{i} is actually in *R[x]*
and is primitive^{(1)}. Thus by Gauss's Lemma each *f*_{i}
is irreducible in *R[x]*. Write *c=a/b* with
*a,b* in *R* and coprime. Then

*bf=af*_{1}...f_{r}

Taking contents^{(1)} of both sides we get *b*content(*f*)=*a*.
Substituting and then cancelling we see that *c* is in *R*
and we have written *f* as a product of irreducibles in
*R[x]*.

Suppose that *f=dg*_{1}...g_{s} is another such product.
Since *k[x]* is a UFD we immediately deduce that
*r=s*. After reordering, if necessary, we have that
*f*_{i}=a_{i}g_{i}, for some *a*_{i}
in *k*. Looking at contents again one deduces that
each *a*_{i} is a unit of *R*, as required.

**Corollary**
Let *K* be a field. Then the polynomial ring in
*n* variables *K[x*_{1},...,x_{n}]
is a UFD.

**Proof:** *K* is a UFD. This gives the base for an induction
with inductive step based on the previous theorem.

Note that *K[x*_{1},...,x_{n}]
is not a principal ideal domain, for *n>=2*,
so this result is nontrivial.
(To see this consider the ideal consisting of all polynomials
with no constant term. If this ideal were to be principal its
generator has to divide both *x*_{1} and *x*_{2}
which are irreducible, so this is clearly impossible.)

(1) The terms **primitive** and **content** are defined
in the Gauss's Lemma writup.