Editing Hilbert's basis theorem (section)

==Statement==
If <math>R</math> is a [[ring (mathematics)|ring]], let <math>R[X]</math> denote the ring of [[polynomial]]s in the indeterminate <math>X</math> over <math>R</math>. [[David Hilbert|Hilbert]] proved that if <math>R</math> is "not too large", in the sense that if <math>R</math> is Noetherian, the same must be true for <math>R[X]</math>. Formally,

<blockquote>'''Hilbert's Basis Theorem.''' If <math>R</math> is a Noetherian ring, then <math>R[X]</math> is a Noetherian ring.<ref>{{harvnb|Roman|2008|loc=p. 136 §5 Theorem 5.9}}</ref></blockquote>
<blockquote>'''Corollary.''' If <math>R</math> is a Noetherian ring, then <math>R[X_1,\dotsc,X_n]</math> is a Noetherian ring.</blockquote>

Hilbert proved the theorem (for the special case of [[multivariate polynomial]]s over a [[field (mathematics)|field]]) in the course of his proof of finite generation of [[ring of invariants|rings of invariants]].{{r|Hilbert1890}} The theorem is interpreted in [[algebraic geometry]] as follows: every [[algebraic set]] is the set of the common [[root of a polynomial|zeros]] of finitely many polynomials. 

Hilbert's proof is highly [[non-constructive proof|non-constructive]]: it proceeds by [[mathematical induction|induction]] on the number of variables, and, at each induction step uses the non-constructive proof for one variable less. Introduced more than eighty years later, [[Gröbner bases]] allow a direct proof that is as constructive as possible: Gröbner bases produce an algorithm for testing whether a polynomial belong to the ideal generated by other polynomials. So, given an infinite sequence of polynomials, one can construct algorithmically the list of those polynomials that do not belong to the ideal generated by the preceding ones. Gröbner basis theory implies that this list is necessarily finite, and is thus a finite basis of the ideal. However, for deciding whether the list is complete, one must consider every element of the infinite sequence, which cannot be done in the finite time allowed to an algorithm.