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  "body": "\n## Statement\n\nLet $R$ be a [[Noetherian|0MVH4Y1CRZ]] [[commutative ring|0GYGHKX834]].\nThen the [[polynomial ring|0Q9EZS90GY]] $R[x]$ is Noetherian.\n\n## Remarks\n\nBy 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.\nThe proof studies the ideals of the leading coefficients of the polynomials of each degree in an ideal of $R[x]$.\nIn algebraic geometry, the theorem says that every set of polynomial equations has the same solutions as a finite subset of it.\n\n## History\n\nDavid Hilbert proved the theorem in 1890, for polynomials in several variables over a field, in an article on invariant theory.\nHis proof was not constructive. It did not give a method to find the finite set of generators.\n\n## Explanations\n\n### L1 human\n\nAn ideal of polynomials can contain infinitely many polynomials.\nThe theorem says that a finite list of them always generates the whole ideal.\nSo every system of polynomial equations can be replaced by a finite one.\n",
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