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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?

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1 Answer 1

up vote 8 down vote accepted

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.

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