How can you determine if a composite extension is purely inseparable?

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The discussion focuses on determining whether the composite extension K1K2/F is purely inseparable, given that K1/F and K2/F are purely inseparable extensions. It is established that if K1 and K2 are generated by purely inseparable elements, then their composite K1K2 is also purely inseparable. The key insight is that if x is an element of K1 and y is an element of K2, then x^p^n and y^p^m are in F for some integers m and n, which supports the conclusion regarding the nature of the composite extension.

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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!
 
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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 xpn and ypm are in F for some m and n. But, how do you use this to show that K1K2/F is a purely inseparable extension?
 
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.
 

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