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

*Theorem:* Every ideal $I$ of $k[x_1, \dots, x_n]$ is finitely generated.
First we need a lemma:
#### § Lemma:

Let $I \subseteq k[x_1, \dots, x_n]$ be an ideal. (1) $(LT(I)) \equiv$ is
a *monomial ideal*. An ideal $I$ is a monomial ideal if there is a subset
$A \subseteq \mathbb Z^n_{\geq 0}$ (possibly infinite) such that $I = (x^a : a \in a)$.
That is, $I$ is generated by monomials of the form $x^a$. Recall that since we
have
#### § Proof of hilbert basis theorem

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

#### § References

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