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It is well known that if a field K is quasi-algebraically closed (i.e. all forms with coefficients in K of degree d in n > d variables have a non-trivial zero in K) then it has no central divison algebras. It is also known that there are no central division algebras over the rational cyclotomic field (obtained by adjoining to Q all roots of unity).

Is it known whether the rational cyclotomic field is quasi-algebraically closed?

For instance, I know that the answer is yes for p-adic cyclotomic fields and that the converse of the first result above is false.

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up vote 4 down vote accepted

This is an old conjecture of Artin and, as far as I know, it is still open. It is mentioned as such on page 477 of this 2007 article on the work of Lang:

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