In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert in connection with the syzygy (relation) problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants , then relations between the relations, and so on, it explains how far one has to go to reach a clarified situation. It is now considered to be an early result of homological algebra, and through the depth concept, be a measure of the non-singularity of affine space.

A contemporary formal statement is the following. Let k be a field and M a finitely generated module over the polynomial ring

.

Hilbert's syzygy theorem then states that any free resolution of M has length at most n.