The Makar-Limanov’s Construction of Algebraically Closed Skew Field via Mal’cev—Newmann Series

Abstract

There is another proof of the L. G. Makar-Limanov’s theorem of existence an algebraically closed skew field in the following sense: every (general) polynomial equation has a root in this field. The example constructed differs from the original one, the Makar-Limanov’s skew field is containing in our example as a subfield. We have used the main ideas of the original proof: the skew field is constructed via Mal’cev—Newmann series. It’s proved that there is an additional property of the skew field. More precisely we have shown an existence of non-zero solutions for every general polynomial equation containing two or more homogeneous components. We consider also the P. Cohn’s definitions of (non-comutative) algebraically closed skew fields and the problems connected.