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?
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? 


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 $\alpha^2 = b^2c^2d^2$, 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. 

