In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert (1890) 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, to be a measure of the non-singularity of affine space. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations by or about: Mathematics Look up Mathematics in Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ... David Hilbert David Hilbert ( January 23, 1862 – February 14, 1943) was a German mathematician born in Wehlau, near Königsberg, Prussia (now Znamensk, near Kaliningrad, Russia) who is recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ... Syzygy can refer to any of several different things: Astronomy In astronomy, a syzygy (Greek: yoked together) is a situation where three celestial bodies are positioned along a straight line. ... In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In commutative algebra, if R is a commutative ring and M an R-module, a R-regular sequence on M is a d-tuple of (non-zero non-unit) elements r1, r2, ..., rd from R such that for each i, ri is not a zerodivisor on the quotient R-module... ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ...

A contemporary formal statement is the following. Let k be a field and M a finitely generated module over the polynomial ring In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ...

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Hilbert's syzygy theorem then states that there exists a free resolution of M of length at most n.