Is it true that any finite dimensional division algebra over a pseudo-algebraically closed field is trivial? We know that this is true for algebraically closed field.



Finite dimensional division algebra over pseudo-algebraic closed field
The fact that every finite dimensional division algebra is trivial is not only true over an algebraically closed field but it is in fact equivalent to the field being algebraically closed. Remember that (extension-)fields are just a special case of division algebras. Thus, if it would be true over pseudo-algebraically closed fields, we would have that a field is algebraically closed if and only if it is pseudo-algebraically closed.  

However, this is not the case, as for example finite fields are pseudo-algebraically closed.


