Hilbert's basis theorem

Statement

Let $R$ be a Noetherian commutative ring. Then the polynomial ring $R[x]$ is Noetherian.

Remarks

By induction, $R[x_1, \dots, x_n]$ is Noetherian for every $n$. In particular, every ideal of $K[x_1, \dots, x_n]$ for a field $K$ is finitely generated. The proof studies the ideals of the leading coefficients of the polynomials of each degree in an ideal of $R[x]$. In algebraic geometry, the theorem says that every set of polynomial equations has the same solutions as a finite subset of it.

History

David Hilbert proved the theorem in 1890, for polynomials in several variables over a field, in an article on invariant theory. His proof was not constructive. It did not give a method to find the finite set of generators.

Explanations

L1 human

An ideal of polynomials can contain infinitely many polynomials. The theorem says that a finite list of them always generates the whole ideal. So every system of polynomial equations can be replaced by a finite one.