Edit: After reading proofs of this theorem, it seems to me that this is similar in spirit to an already established proof (Proof 2 on ProofWiki, for instance).
This is a constructive proof of Hilbert’s Basis Theorem.
Hilbert’s Basis Theorem says that if
is a Noetherian ring (every ideal has a finite number of generators), then so is the polynomial ring
.
Let
be an ideal. It contains polynomials and constants. Let us take the set of all leading coefficients of the polynomials in
, and call it
. This is clearly an ideal! Hence,
. There are polynomials of the form
in
.
Let
be
.
Now take the set
. The **leading coefficients** of polynomials of degree
for each
,
, form an ideal. Hence, there must be a finite number of generators for the ideal containing leading coefficients of polynomials of degree
.
For example, take all polynomials of degree
in
. The leading coefficients of such polynomials
form an ideal (try adding these polynomials or multiplying them with elements from
). As
is Noetherian, the ideal of leading coefficients has a finite number of generators- say
. It follows that
belong to this ideal of polynomials of degree
, and can generate the leading coefficient of **any** polynomial of degree
in
.
Proceeding by recursion, we will soon have the finite list of generators generating the leading coefficients of all polynomials of degree
. We can hence say that we have the list of polynomials that can be linearly added to generate all polynomials of degree
. This is not difficult to see. It is important to note that we can also linearly generate the polynomials with leading coefficients
.
Important note on notation: let
be the finite number of coefficients which generate the leading coefficients of all polynomials of degree
belonging to
.
It is easy to see that
generates the whole of
. For polynomials of degree
, we’ve already shown how. For polynomials of degree
, generate all leading coefficients using
(which in turn are generated by
), and then generate the rest using
right away.