Let $ K$ be a field and let $ X$ be a set. Denote by $ \mathcal D_K(X)$ the free skew $ K$ -field on $ X$ .

Assume that $ |X|\ne |Y|$ . Is it true that $ \mathcal D_K(X)$ and $ \mathcal D_K(Y)$ are not isomorphic?

There are several constructions of $ \mathcal D_K(X)$ . The standard one is the following. Let $ F_X$ be a free group on $ X$ . Fix a bi-invariant total order $ \le$ on $ F_X$ and let $ K_{\le}((F_X))$ be the Malcev-Neumann ring of formal series (it consists of formal series having well-ordered support). One proves that $ K_{\le}((F_X))$ is a skew field. Then $ \mathcal D_K(X)$ is isomorphic to the division closure of the group algebra $ K[F_X]$ in $ K_{\le}((F_X))$ .