The question says:
Find the degrees of the splitting extensions of the following polynomials, and show that
in each case the number of automorphisms of the splitting field is at most the degree
of the extension.
i) x^3 - 1 over Q
the fact that if f is irreducible over K with splitting field L, and a,b are roots of f, then K(a) is isomorphic to K(b), so there is an automorphism i:K -> K with i(a) = b
The Attempt at a Solution
Just a bit confused by the question- does it mean the number of automorphisms that fix the original field K?? I'm fine with finding the degree of the extension its the second bit thats new to me.
I *think* for the example shown, the degree of the extension is 2 and there is only one automorphism, that taking exp(i*PI/3) to exp(-i*PI/3). Clearly 1 is less than or equal to 2 but I'm not sure a) if this is right or b) how to generalise this concept to the other examples.