Roots of unity and field extensions

Question

Can we always break an arbitrary field extension $L/K$ into an extension $F/K$ in which the only roots of unity of $F$ are those in $K$, followed by an extension $L/F$ which is of the form $L=F(\{\omega_i\})$ where the $\omega_i$ are roots of unity? What if we restrict to $L/K$ separable, or finite?

Answer 1

The question has been answered on MathOverflow: http://mathoverflow.net/questions/49913/factoring-a-field-extension-into-one-which-adds-no-roots-of-unity-followed-by-on/49914#49914