Let be an extension of such that where is transcedental over and satisfies the equation over . Find the Galois group of over for any Show that

is contained in . Use this to prove that the function cos is expressible rationally with complex coefficients in terms of and . Does this hold for sin .

, we must have

(1)

(2)

(3)

This means . So there exist , a n-th root of 1, such that . Similarly such that . (3) becomes

. Combing with , we get . Since , we know that when is even and when is odd and .

In conclusion, the Galois group is trivial when is odd and Klein group when is even.

No matter is even and odd, is invariant, so is in .

If is odd, is in . If is even, is not in the ground field, because can not fix .

At first I am surprised by the result I got. I can not believe that the galois is trivial when is odd. So I conculted to Zhuohui. Thank him for his kind help. I drew the following picture after discussion with him.