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Hilbert's basis theorem ：ウィキペディア英語版

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

R

a ring, let

R()

denote the ring of polynomials in the indeterminate

X

over

R

. Hilbert proved that if

R

is "not too large", in the sense that if

R

is Noetherian, the same must be true for

R()

. Formally,

Hilbert's Basis Theorem. If $R$ is a Noetherian ring, then $R()$ is a Noetherian ring.

Corollary. If $R$ is a Noetherian ring, then $R()$ 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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