Hilbert's basis theorem
Statement

If R {\displaystyle R} is a ring, let R {\displaystyle R} denote the ring of polynomials in the indeterminate X {\displaystyle X} over R {\displaystyle R}. Hilbert proved that if R {\displaystyle R} is "not too large", in the sense that if R {\displaystyle R} is Noetherian, the same must be true for R {\displaystyle R}. Formally,

Hilbert's Basis Theorem. If R {\displaystyle R} is a Noetherian ring, then R {\displaystyle R} is a Noetherian ring.

Corollary. If R {\displaystyle R} is a Noetherian ring, then R {\displaystyle R} is a Noetherian ring.

Hilbert proved the theorem (for the special case of multivariate polynomials over a field) in the course of his proof of finite generation of rings of invariants. The theorem is interpreted in algebraic geometry as follows: every algebraic set is the set of the common zeros of finitely many polynomials.

Hilbert's proof is highly non-constructive: it proceeds by induction on the number of variables, and, at each induction step use the non-constructive proof for one variable less. Introduced more 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.
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