Great, I thought there must be a simple counterexample. To come full circle to the supposed proof, since the minimum polynomial of ##\sqrt[4]{2}## in ##I[x]## is ##x^4-2## and this is reducible in ##E[x]## as ##(x^2+\sqrt{2})(x^2-\sqrt{2})##, this is where the proof fails.
And the other proof...
But what if ##K## and ##K'## are "different," i.e. isomorphic but not equal, spitting fields? Isn't a map ##h:\mathbb{Q}(\sqrt{2})\rightarrow \mathbb{Q}[x]/\langle x^2-2\rangle## fixing ##\mathbb{Q}## and matching roots of ##x^2-2## a counterexample?
Or, taking a different approach, if we...
He uses a result from the previous page that ##h## takes elements of ##K## to elements of ##K##, but this was when he assumed a common extension at the start, which he doesn't do in the proof. I think he must have meant to but messed up when rewriting the section for the second edition.
Pinter...
I'm self-studying A Book of Abstract Algebra, 2nd ed, by Pinter and I have two questions. First, the author says to consider the situation where ##K## and ##K'## are finite extensions of ##F##, and furthermore that ##K## and ##K'## have a common extension ##E##. Then he goes on to prove that if...