The Makar-Limanov algebraically closed skew field

We re-prove the Makar-Limanov theorem on the existence of an algebraically closed skew field in the sense of there being a solution for any (generalized) polynomial equation. A new example of such a skew field is presented in which the Makar-Limanov construction is contained as a skew subfield. Our reasoning is underpinned by the main ideas of the original proof, but we employ a simpler argument for proving that the skew field constructed is algebraically closed.