Unique factorization in the polynomial
ring in n
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].
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
for c in k and each fi irreducible
in k[x]. By adjusting c we can arrange
that each fi is actually in R[x]
and is primitive(1). Thus by Gauss's Lemma each fi
is irreducible in R[x]. Write c=a/b with
a,b in R and coprime. Then
Taking contents(1) of both sides we get bcontent(f)=a.
Substituting and then cancelling we see that c is in R
and we have written f as a product of irreducibles in
Suppose that f=dg1...gs is another such product.
Since k[x] is a UFD we immediately deduce that
r=s. After reordering, if necessary, we have that
fi=aigi, for some ai
in k. Looking at contents again one deduces that
each ai is a unit of R, as required.
Let K be a field. Then the polynomial ring in
n variables K[x1,...,xn]
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[x1,...,xn]
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 x1 and x2
which are irreducible, so this is clearly impossible.)
(1) The terms primitive and content are defined
in the Gauss's Lemma writup.