---
tag: B1Q5F5VNSK
kind: problem
name: Kurosh problem
msc: [16N40, 16P90, 16R10]
uses: [3CTW1P4HBP, 04C5ZBG1WT, 0H628RSGWX, NZ1G9WRWFB, RC5A5T7YY2, C0EDT81EGH, XWEMJ7T2XV, 0EA5BTA0CX]
about: [3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Kurosh_problem
same_as:
  wikidata: Q6446395
  wikipedia: Kurosh problem
history:
  posed_by: "Aleksandr Gennadievich Kurosh"
  first_stated: "1941"
---

## Problem

Let $k$ be a [[field|04C5ZBG1WT]] and let $A$ be an [[associative algebra|3CTW1P4HBP]] over $k$ that is generated by finitely many elements.
Assume that $A$ is algebraic: every element of $A$ is [[algebraic|0H628RSGWX]] over $k$, that is, a root of a nonzero polynomial with coefficients in $k$.
Must $A$ be finite-dimensional over $k$?

## Remarks

The problem is a ring-theoretic analogue of the [[Burnside problem|XWEMJ7T2XV]] for periodic groups.
An important special case asks whether a finitely generated nil algebra, in which every element is nilpotent, must be nilpotent and hence finite-dimensional.

The answer is negative in general.
Golod used the [[Golod–Shafarevich theorem|RC5A5T7YY2]] in 1964 to construct a finitely generated infinite-dimensional nil algebra.
The same construction gave infinite finitely generated periodic groups, a negative answer to the general Burnside problem.

The answer is positive for algebras that satisfy a polynomial identity: a finitely generated algebraic [[polynomial identity algebra|C0EDT81EGH]] is finite-dimensional.
In particular it is positive when the elements of $A$ are algebraic of bounded degree.

A variant asks the same question for [[division rings|0EA5BTA0CX]] that are algebraic over their centre and finitely generated over it.

## Explanations

### L1 human

Each element of an algebraic algebra satisfies some polynomial equation, so each single element generates only a finite-dimensional piece.
Kurosh asked whether finitely many such elements must then generate a finite-dimensional algebra.
Surprisingly, the answer is no: there are algebras generated by a few elements, each element is even nilpotent, and yet the algebra is infinite-dimensional.