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Let $k$ be a field, $D$ be a central division algebra of degree $n$ over $k$. We call $k'$ a splitting field of $D$ if $D\otimes_kk'\cong M_n(k')$. Splitting fields may not be isomorphic, can we say more about them?

Let $L_1,L_2$ be splitting fields of $D/k$, do we know if $\mathrm{Gal}(L_i/k)$ are isomorphic?

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  • $\begingroup$ See here what class field theory says about it (when $k$ is a number field). $\endgroup$ – Jyrki Lahtonen Jul 8 '18 at 15:13
  • $\begingroup$ @JyrkiLahtonen If $k=\mathbb{Q}$, let $D/k$ be the division algebra of degree $n$. Suppose $L$ splits $D$, then $[L:k]=n$, $[L_v:k_p]\mathrm{inv}_p(D)=0$. (Sorry I have a stupid question, isn't $[L:k]=[L_v:k_p]$ since completion doesn't change dimension) $\endgroup$ – Qixiao Jul 8 '18 at 15:58
  • $\begingroup$ Completion will change dimension when the prime $p$ splits in $L/k$. See here. $\endgroup$ – Jyrki Lahtonen Jul 8 '18 at 16:16
  • $\begingroup$ @JyrkiLahtonen Do you think the following is an example: Let $L/k$ be a noncyclic Galois extension, take a division algebra $D$ in $\mathrm{Br}(L/K)$, then $L$ is a maximal subfield of $D$, but on the other hand, every division algebra over number field is cyclic, so some cyclic $L'/k$ is also a subfield of $D$. $\endgroup$ – Qixiao Jul 8 '18 at 17:06
  • $\begingroup$ Also would the maximal subfield be unique up to isomorphism if we add some other conditions as unramified? $\endgroup$ – Qixiao Jul 8 '18 at 17:12
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Let me suppose $k$ is a number field, and $D/k$ is a central division algebra. If $K/k$ is a number field embedding in $D$, then $K$ splits $D$ if and only if $K$ is maximal. Any field containing such a $K$ will also split $D$.

Say $n$ is the degree of $D$, i.e., $n^2 = dim_k D$. Then the maximal subfields of are the extensions of $k$ of degree $n$ which have suitable splitting behavior at any prime where $D$ is ramified. E.g., suppose $D_v$ is division at any place it is ramified. Then a degree $n$ field extension $K$ of $k$ embeds in $D$ if and only it it is not split ($K_v$ is a field) at all $v$ ramifying in $D$.

If $K_1$ and $K_2$ are maximal subfields of $D$, then you can only guarantee their Galois groups over $k$ are isomorphic when $n \le 2$.

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