Proving Abel's theorem of 'natural irrationalities' using Galois Theory
In Ian Stewart's book "Galois Theory", there's a discussion on Abel and Ruffini's original work on the general quintic, where he explains the gap in Ruffini's proof which was subsequently fixed by Abel with his theorem on "Natural Irrationalities".
Let be independent complex variables, the corresponding elementary symmetric polynomials, , (where C is the field of complex numbers), so that the general polynomial splits over the extension .
Abel's theorem on Natural Irrationalities is then stated as follows:
If contains an element that lies in some radical extension , then there exists a radical extension with and
Stewart proceeds to give Abel's original proof of this, but remarks that a proof using Galois theory is straightforward, and indeed sets this as an exercise much later on in the book. So what I'm having trouble with is just that: proving the above theorem in a more straightforward manner using the Galois correspondence.
Any help would be greatly appreciated...apparently this should be 'straightforward' but I've only made myself more confused trying to work it out. Several other texts refer to a "Theorem of Natural Irrationalities", but these are all different in nature, and I've been unable to translate them back into a proof of the above statement.