I am currently reading some introductory material on Brauer groups ("Noncommutative Algebra", by Farb and Dennis) and the following two questions came to my mind:
1) Are all crossed products algebras, division algebras?
2) Are all division algebras, crossed product algebras?
In chapter 4 the authors state that the answer to question 2 is affirmative in the case of central division algebras over algebraic number fields. I would appreciate an answer to those two questions or maybe a counterexample (if available that is) or even a reference. I should mention that crossed products in the context of Dennis and Farb are associative algebras. Thanks!