Let D be a division algebra over F, E its maximal field. Is it true that:


1) all such fields are equivalent over F?


2) all such fields are conjugate by inner automorphisms of D?



equivalence of maximal fields in division algebras
No, this is very untrue. Let $F=\mathbb{Q}$ and let $D$ be the quaternions with $\mathbb{Q}$ coefficients. For any nonzero $\alpha = bi+cj+dk$ in $D$, $\mathbb{Q}(\alpha)$ is a maximal subfield of $D$. Since $\alpha^2 = -b^2-c^2-d^2$, we can achieve the field $\mathbb{Q}(\sqrt{-D})$ for any $D$ not of the form $4^n (8m+7)$. For example, $\alpha=i$ generates a subfield isomorphic to $\mathbb{Q}(\sqrt{-1})$, $\alpha = i+j$ generates a subfield isomorphic to $\mathbb{Q}(\sqrt{-2})$, $\alpha = i+j+k$ generates a subfield isomorphic to $\mathbb{Q}(\sqrt{-3})$ etc.


