Let $K$ be a field and $D$ be a central division algebra over $K$ of degree $n$. Suppose that $L\subset D$ is a maximal subfield, so that $[L:K]=n$. Then we know that $L$ is a splitting field, so there exists an isomorphism $f:D\otimes\_KL\to M\_n(L)$ of $L$-algebras.


My question is: does it always exists an $f$ as above such that $f(L\otimes \_KL)$ is the set of diagonal matrices in $M\_n(L)$?


(asked in <https://math.stackexchange.com/q/1619116/217671> but I think it might be more appropriate here).



Central division algebras and splitting fields
The answer is "yes" if the field extension $L/K$ is Galois : Anton's proof works in that case.


Classical examples where there exists a Galois degree $n$ extension $L/K$ embedded in $D$ are those of global fields (e.g. numbers fields) and local fields (completions of global fields). In this case the division algebras are moreover cyclic : one may choose $L/K$ cyclic. 


In the case of a cyclic division algebra, you may even choose an explicit isomorphism $f$. Indeed take $L/K$ cyclic with generator $\sigma\in {\rm Gal}(L/K)$. By Skölem-Noether theorem, there exists $x\in D$, well defined up to a right factor in $L^\times$ such that $\sigma (l)=xlx^{-1}$, $l\in L$.


First considering $D$ as a right $L$-vector space, you have a canonical isomorphism of $L$-algebras:


$$
D\otimes\_K L \simeq {\rm End}\_L (D), \ d\otimes l \mapsto \{ u\mapsto dul\} 
$$ 


The family $(1,x,x^2 ,...,x^{n-1})$ is a basis of $D/L$. In this basis, the image of $L\otimes\_K L$ takes a diagonal form. 


