Hi, Let $A$ be a division algebra over a local field $F$ of dimension $n^2$ and $K$ be an extension of $F$ of degree $n$. Then if follows from COROLLARY 2 in page 225 of Weil's Basic Number Theory that $A$ is split over $K$. Now my question is 


1. Does $A$ contain a subfield isomorphic to $K$ ? Why?
2. Could you descirbe the general picture about "maximal subfields in central simple algebras" over a (not necessarily local) field ? Or tell me some references.


Thank you !



Maximal subfields in a division algebra over a local field
Corollaries 3.4 and 3.7 on pages 130 and 131 of Milne's notes (jmilne.org/math/CourseNotes/CFT.pdf) will give you the answers.


