## Hilbert’s Basis Theorem for polynomial rings: Every ideal in a polynomial ring over a field is finitely generated

Let $F$ be a field and let $R = F[x_1, \ldots, x_t]$. Let $I \subseteq R$ be a nonzero ideal. Fix a monomial order. Prove that $I$ has a Gröbner basis with respect to the monomial order and conclude that $I$ is finitely generated.

Let $I \subseteq R$ be a nonzero ideal, and consider the monomial ideal $LT(I)$. By Dickson’s lemma, $LT(I)$ is finitely generated by monomials; say $LT(I) = (a_1, \ldots, a_k)$. Without loss of generality we may assume that no $a_i$ divides another. Recall that $LT(I) = (LT(g_i) \ |\ g_i \in I)$. In particular, we have that for each $i \in [1,k]$, there exist $g_i \in I$ and $j \in [1,k]$ such that $a_j|LT(g_i)|a_i$. In fact, $a_j = a_i$, so that $a_i$ and $LT(g_i)$ are associates. That is, we have $LT(I) = (LT(g_1), \ldots, LT(g_k))$ for some $g_i \in I$. By Proposition 24 on page 322 of D&F, $G = \{g_1, \ldots, g_k\}$ is a Gröbner basis for $I$. Then $I = (G)$ is finitely generated.