$\displaystyle k(x)$ is an automorphism defined by $\displaystyle \sigma_1(x)=x, \sigma_2(x)=1-x$, and $\displaystyle \sigma_3(x)=1/x$ Let $\displaystyle I(x)=\frac{(x^2-x+1)^3}{x^2(x-1)^2}$

Assuming the fundamental theorem of Galois Theory, exhibit all intermediate fields between E=k(x) and F=k(I).