I am seeking resolution of a disagreement over the nature of Galois theory. In the followup comments on a separate thread, I made the claim that the beauty of Galois theory is that it does not require us to distinguish between the various choices for square roots and and cube roots. Another correspondent disagreed; replying to my example, he wrote:
"You would be unable to even define "the Galois structure of x^10−1=0" if radicals did not define functions."
By which I am understanding him to say that in order to do Galois theory, we must have a convention whereby one choice the radical is distinguished as special.
I would claim that Galois theory is precisely about all the things you can say about algebraic structures without having to make any arbitrary choices as to which particular values of a radical we are talking about. To be sure, there are sometimes choices made, but they are never arbitrary. There is no need to have a single-valued function for the square root or the cube root. I wonder if people will agree with this characterization.