Hey, I have been working on this problem for a while now, but I just can't solve it, so I'm posting it to see if anyone has any idea how to work it out:

Let k be an arbitrary field and K=k(x) be the fraction field of the polynomial ring in one variable. Define F,G:K--->K by F(f(x)/g(x))=f(1/x)/g(1/x) and G(f(x)/g(x))=f(1-x)/g(1-x).

1) Find the fixed field L of {F,G}

2)Determine Gal(K/L) (the Galois group of the extension)

3)Find an h in L such that L=k(h)

Thanks in advance.