E and F are free over K if every subset X of E that is algebraically independent over K is also algebraically independent over F.

(a) The definition is symmetric (that is, E and Fare free over K if and only if F and E are free over K).

(b) If E and F are linearly disjoint over K, then E and Fare free over K. Show by example that the converse is false.

(c) If E is separable over K and E and Fare free over K, then EF is separable over F.

(d) If E and F are free over K and both separable over K, then EF is separable over K.

E and Fare always extension fields of a field K, and C is an algebraically closed field containing E and F.