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.