Original proof of Hilbert's syzygy theorem

See Theory of Algebraic Invariants

Actually, there is an English translation of Hilbert's "Über die Theorie der algebraischen Formen" (Mathematische Annalen 36, 473--530, 1890), where the theorem is in Part III of that five-part paper. The translation is in pages 143--224 of "Hilbert's Invariant Theory Papers", Volume VIII of R. Hermann's "Lie Groups: History, Frontiers and Applications", Math. Sci. Press, Brookline, MA, 1978. I don't know if the book is available on-line.