§ Hilbert basis theorem for polynomial rings over fields (TODO)
Theorem: Every ideal of is finitely generated.
First we need a lemma:
Let be an ideal. (1) is
a monomial ideal. An ideal is a monomial ideal if there is a subset
(possibly infinite) such that .
That is, is generated by monomials of the form . Recall that since we
§ Proof of hilbert basis theorem
- We wish to show that every ideal of is finitely generated.
- If then take and we are done.
- Pick polynomials such that .This is always possible from our lemma.We claim that .
- Since each , it is clear that .
- Conversely, let be a polynomial.
- Divide by to get where no termof is divisible by . We claim that .
- Cox, Little, O'Shea: computational AG.