Galois extensions inside a division ring

Question

Let $D$ be a division ring which has finite dimension over its centre.

Q1. Under which conditions can one find a maximal subfield $K$ of $D$ and a proper subfield $L$ of $K$ such that $K/L$ is Galois?

Q2. Is it possible that such an $L$ does not exist ?

These questions are motivated by Dickson's construction of cyclic algebra, constructing central simple algebra starting from any cyclic Galois extension. I am looking for a kind of converse of this construction, wondering to what extent a simple central algebra (particularly a division ring finite dimensional over its centre) is built up with a cyclic Galois extension.

Edit. Thanks to LSpice's comments, the questions reduces to:

Q3. Under which conditions can one find a maximal subfield $L\subset D$ whose normalizer in $D$ properly contains $L$ ?

Answer 1

I'm not sure if it is relevant to answer more than a year after the question was raised, especially because I'm not really fully answering the question.

First, I would like to point out that an answer to the question would give a positive answer to the very hard and still open following question: let $p$ be a prime number, and let $F$ be a field of characteristic different from $p$. Is any central division $F$-algebra of degree $p$ cyclic (i.e. does it has a cyclic maximal subfield)? This is known for $p=2,3$...and that's it. No one knows what is the correct answer even for $p=5.$

Now assume that $D$ is a division ring with center $F$, which is of degree $p$ over $F.$ Any maximal subfield $L$ containing $F$ has degree $p$, so the only proper subfield of $L$ is $F$. Asking if one can choose $L/F$ to be Galois is then equivalent to ask whether $L/F$ can be chosen to be cyclic, which is exactly the statement of the previous question.

On the positive side, one can show that if $D$ is a central division $F$-algebra of degree $p^m$ ($p$ prime), then the answer is "yes" up to a prime-to-$p$ extension, if $F$ has not characteristic $p$:

By a well-known theorem of Albert, one can find a prime-to-$p$ extension $E/F$ such that $D\otimes_F E$ (which is still division) contains a maximal subfield $L/E$ such that there is a filtration $F=L_0\subset L_1\subset\cdots \subset L_{r-1}\subset L_r=L$, where each $L_i/L_{i-1}$ has degree $p$. Now $E(\zeta_p)/E$ is prime to $p$, so replacing $E$ by $E(\zeta_p)$, one may assume that $E$ contains $\zeta_p$. In this case , $L/L_{r-1}$ is a cyclic Kummer extension.

To finish with this long message, I just don't understand why @Drike wants a Galois extension "from the above". Even if the answer to the question is "yes", I don't see what interesting result it could imply on the structure of central division algebras. From my point of view, the structure of subfields of division algebras are more interesting. Just to remind the potential future readers of this text: a central division $F$-algebra does not always contain a Galois maximal subfield (there are counter examples in degree $8$).