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

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To determine if the composite extension K1K2/F is purely inseparable given that K1/F and K2/F are purely inseparable, one can use the property that if x is in K1 and y is in K2, then x^p^n and y^p^m belong to F for some integers m and n. This implies that the elements of K1K2 can be expressed in terms of elements from K1 and K2, maintaining the purely inseparable nature. The argument hinges on the fact that K1K2 is generated by the union of the elements from K1 and K2. Thus, if both extensions are purely inseparable, their composite extension K1K2/F will also be purely inseparable. This conclusion follows from the definition that an algebraic extension is purely inseparable if it is generated by purely inseparable elements.
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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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