# Free skew fileds over sets of different cardinal

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 $$K_{\le}((F_X))$$.