Problem: If is separable over field , where is an extension of , and , then . If , then .

It is clear to me that in characteristic 0 this proof is trivial. In characteristic , with prime, one direction of the inclusion is obvious for each case, and that is . To get the other direction I was thinking of using the identity , where is the power that makes . Therefore, we must have that by subtracting off (since it is an element of ). If is an element of , then we can subtract it off of the element , and the reverse containment will follow. If , then it is purely inseparable over . Similar reasoning can be used to show is purely inseparable over (otherwise would be an element of and we'd be done), hence is a purely inseparable extension of .

My issue is that I have no idea where to go with this; I'm hoping to find some kind of contradiction, but I am probably lacking somewhere in my understanding of what it means for an element to be separable. Any ideas? Is my approach even close to correct? I've been on this problem for 2 days, with no luck.