Concept

# Hilbert's basis theorem

Summary
In mathematics, specifically commutative algebra, Hilbert's basis theorem says that a polynomial ring over a Noetherian ring is Noetherian. Statement If R is a ring, let R[X] 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[X]. Formally, Hilbert's Basis Theorem. If R is a Noetherian ring, then R[X] is a Noetherian ring. Corollary. If R is a Noetherian ring, then R[X_1,\dotsc,X_n] 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. Hilbert proved the theorem (for the special case of polynomial rings over a field) in the course of his proof of finite generation of rings
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