Purely Inseparable Extensions

[kavoukoff1]

Could anyone give me some help for showing if K1/F and K2/F are purely inseparable extensions, then K1K2/F is purely inseparable. Thanks!

 

[kavoukoff1]

Sorry, I forgot to put up my thoughts/attempts at this problem. Do I use the fact that if x is an element of K1 and y is an element of K2, then x^pn and y^pm are in F for some m and n. But, how do you use this to show that K1K2/F is a purely inseparable extension?

 

[Landau]

This is not my area of expertise, but here's a thought:

If E/K is an algebraic extension, then it is purely inseparable iff it is generated by purely inseparable elements.

K1K2/F is generated by the union of the elements by which K1 and K2 are generated.