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

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.