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Hilbert's basis theorem
In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian.
==Statement==
If a ring, let denote the ring of polynomials in the indeterminate over . Hilbert proved that if is "not too large", in the sense that if is Noetherian, the same must be true for . Formally,
Hilbert's Basis Theorem. If is a Noetherian ring, then is a Noetherian ring.
Corollary. If is a Noetherian ring, then is a Noetherian ring.
This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the set of common roots of finitely many polynomial equations. proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings of invariants.
Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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