Different Definitions of Algebraically Closed Skew Fields

We consider an algebraically closed (in the sense of solvability of arbitrary polynomial equations) skew field constructed by Makar – Limanov. It is shown that every generalized polynomial equation with more than one homogeneous component has a non-zero solution. We also look into P. Cohn's approach to defining algebraically closed non-commutative skew fields and treat some related problems.