Let be a field and let . Let be a nonzero ideal. Fix a monomial order. Prove that has a Gröbner basis with respect to the monomial order and conclude that is finitely generated.
Let be a nonzero ideal, and consider the monomial ideal . By Dickson’s lemma, is finitely generated by monomials; say . Without loss of generality we may assume that no divides another. Recall that . In particular, we have that for each , there exist and such that . In fact, , so that and are associates. That is, we have for some . By Proposition 24 on page 322 of D&F, is a Gröbner basis for . Then is finitely generated.