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edited Apr 13, 2017 at 12:58

I'm not sure exactly which kind of proof you're looking for, but a proof by induction using Gröbner bases is presented in Chapter 15 of Eisenbud's Commutative Algebra, see Corollary 15.11 specifically.

There's also a more abstract homological proof in Chapter 19, see this related question: Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan

The first proof is closer in spirit with Hilbert's original proof.

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created Jun 11, 2010 at 19:50

Steven Sam

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There's also a more abstract homological proof in Chapter 19, see this related question: Best exposition of the Proof of the Hilbert Syzygy Theorem by Eilenberg-Cartan

The first proof is closer in spirit with Hilbert's original proof.