Hilbert's Basis Theorem
Hilbert's Basis Theorem is a result concerning Noetherian
rings. It states that if is a (not necessarily commutative)
Noetherian ring, then the ring of polynomials
is also a Noetherian ring. (The converse is
evidently true as well.)
Note that must be finite; if we adjoin infinitely many
variables, then the ideal generated by these variables
is not finitely generated.
The theorem is named for David Hilbert, one of the great mathematicians of the late nineteenth and twentieth centuries. He first stated and proved the theorem in 1888, using a nonconstructive proof that led Paul Gordan to declare famously, "Das ist nicht Mathematik. Das ist Theologie. [This is not mathematics. This is theology.]" In time, though, the value of nonconstructive proofs was more widely recognized.
Proof
By induction, it suffices to show that if is a Noetherian
ring, then so is
.
To this end, suppose that
is an
ascending chain of (two-sided) ideals of
.
Let denote the set of elements
of
such that there is a polynomial in
with
degree
and with
as the coefficient of
. Then
is a two-sided ideal of
; furthermore,
for any
,
,
Since
is Noetherian, it follows that for every
,
the chain
stabilizes to some ideal
. Furthermore, the
ascending chain
also stabilizes to some ideal
.
Then for any
and any
,
We claim that the chain
stabilizes
at
. For this, it suffices to show that for
all
,
. We will
thus prove that all polynomials of degree
in
are also elements of
, by induction on
.
For our base case, we note that , and these ideals are the sets of degree-zero
polynomials in
and
, respectively.
Now, suppose that all of 's elements of degree
or lower are also elements of
. Let
be an element of degree
in
. Since
there exists some element
with the same
leading coefficient as
. Then by inductive hypothesis,
so
as desired.