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.

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  • Ramón Pino Pérez  On November 20, 2011 at 7:39 pm

    Thanks very much for your work. That is very useful.
    By the way, what is the reference D&F?
    Best regards,
    Ramón Pino Pérez

    • nbloomf  On November 21, 2011 at 11:32 am

      D&F is the incredibly useful book ‘Abstract Algebra’ by Dummit and Foote.

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