§ Hilbert basis theorem for polynomial rings over fields (TODO)

Theorem: Every ideal II of k[x1,,xn]k[x_1, \dots, x_n] is finitely generated. First we need a lemma:

§ Lemma:

Let Ik[x1,,xn]I \subseteq k[x_1, \dots, x_n] be an ideal. (1) (LT(I))(LT(I)) \equiv is a monomial ideal. An ideal II is a monomial ideal if there is a subset AZ0nA \subseteq \mathbb Z^n_{\geq 0} (possibly infinite) such that I=(xa:aa)I = (x^a : a \in a). That is, II is generated by monomials of the form xax^a. Recall that since we have

§ Proof of hilbert basis theorem

  • We wish to show that every ideal II of k[x1,,xn]k[x_1, \dots, x_n] is finitely generated.
  • If I={0}I = \{ 0 \} then take I=(0)I = (0) and we are done.
  • Pick polynomials gig_i such that (LT(I))=(LT(g1),LT(g2),,LT(gt))(LT(I)) = (LT(g_1), LT(g_2), \dots, LT(g_t)).This is always possible from our lemma.We claim that I=(g1,g2,,gt)I = (g_1, g_2, \dots, g_t).
  • Since each giIg_i \in I, it is clear that (g1,,gt)I(g_1, \dots, g_t) \subseteq I.
  • Conversely, let fIf \in I be a polynomial.
  • Divide ff by g1,,gtg_1, \dots, g_t to get f=iaigi+rf = \sum_i a_i g_i + r where no termof rr is divisible by LT(g1),,LT(gt)LT(g_1), \dots, LT(g_t). We claim that r=0r = 0.

§ References

  • Cox, Little, O'Shea: computational AG.