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Free subalgebras of division algebras over uncountable fields

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We study the existence of free subalgebras in division algebras, and prove the following general result: if A is a noetherian domain which is countably generated over an uncountable algebraically closed field k of characteristic 0, then either the quotient division algebra of A contains a free algebra on two generators, or it is left algebraic over every maximal subfield. As an application, we prove that if k is an uncountable algebraically closed field and A is a finitely generated k-algebra that is a domain of GK-dimension strictly less than 3, then either A satisfies a polynomial identity, or the quotient division algebra of A contains a free k-algebra on two generators. © 2014 Springer-Verlag Berlin Heidelberg.

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