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  "body": "\n## Problem\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ that is generated by finitely many elements.\nAssume that $A$ is algebraic: every element of $A$ is [[algebraic|0H628RSGWX]] over $k$, that is, a root of a nonzero polynomial with coefficients in $k$.\nMust $A$ be finite-dimensional over $k$?\n\n## Remarks\n\nThe problem is a ring-theoretic analogue of the [[Burnside problem|XWEMJ7T2XV]] for periodic groups.\nAn important special case asks whether a finitely generated nil algebra, in which every element is nilpotent, must be nilpotent and hence finite-dimensional.\n\nThe answer is negative in general.\nGolod used the [[Golod–Shafarevich theorem|RC5A5T7YY2]] in 1964 to construct a finitely generated infinite-dimensional nil algebra.\nThe same construction gave infinite finitely generated periodic groups, a negative answer to the general Burnside problem.\n\nThe answer is positive for algebras that satisfy a polynomial identity: a finitely generated algebraic [[polynomial identity algebra|C0EDT81EGH]] is finite-dimensional.\nIn particular it is positive when the elements of $A$ are algebraic of bounded degree.\n\nA variant asks the same question for [[division rings|0EA5BTA0CX]] that are algebraic over their centre and finitely generated over it.\n\n## Explanations\n\n### L1 human\n\nEach element of an algebraic algebra satisfies some polynomial equation, so each single element generates only a finite-dimensional piece.\nKurosh asked whether finitely many such elements must then generate a finite-dimensional algebra.\nSurprisingly, the answer is no: there are algebras generated by a few elements, each element is even nilpotent, and yet the algebra is infinite-dimensional.\n",
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      "posed_by": "Aleksandr Gennadievich Kurosh"
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