Consider the automorphisms of k(x) defined by s1(x) = x (the

identity), s2(x) = 1 - x and s3(x) = 1/x. Show that these generate a group

G of order exactly 6 and that the remaining automorphisms are s4(x) = 1 -

(1/x); s5(x) = 1/(1 - x), and s6(x) = x/(x - 1).

Assuming the fundamental theorem of the Galois theory, exhibit

all the intermediate fields between E = k(x) and F = k(I).

I(x)= (x^2-x+1)^3 / (x^2*(x-1)^2)

Also Show I(x) is invariant under the group G