Let F be a field, and E/F an infinite algebraic extension. Let D be a finite dimensional division algebra over E (meaning its center is also E).

Is it possible to somehow gow down to a finite dimensional division algebra over a finite algebraic extension of F? More precisely, does there exist a finite dimensional division algebra D' over K, where K/F is finite algebraic and $K \subseteq E$, such that $D' \otimes_K E= D$? What about the relationship between the degrees of $D$ and $D'$?

The naive approach I have tried: Let $m_1, \dots, m_n$ be a basis of D over E. Let K be a finite field extension of F containing all the elements of E needed to write the products and inverses of $m_1, m_2, \dots, m_n.$ Take the finite dimensional K-algebra generated by the $m_i.$ This is where I get stuck, because I don't see why this would be a division algebra.

Any ideas or references would be very helpful.