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Is there a division algebra $D$ with center $K$ that satisfies the following 3 conditions?

1) $D$ is of infinite dimension over $K$;

2) every element of $D$ is algebraic over $K$;

3) $D$ is finitely generated (as division $K$-algebra).

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This is a fairly well known old and open (as far as I know) problem: Kurosh’s Problem for division rings. See, for example, Question 3 in Agata Smoktunovicz’s 2006 ICM talk.

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