Prove or disprove Two different subgroups of a Galois group

Prove or disprove: Two different subgroups of a Galois group will have different fixed fields.

Solution

Let L/K be a Galois extension with the Galois group G.

Let S and T be two distinct subgroups of G.

Then U and V be the fixed fields of S and T.

Then Gal (L/U ) = S and Gal(L/V)=T.

If S fixes V then S is a subgroup of T and conversely if T fixes U then T is a subgroup of S.

As S and T are distinct, this implies that the fixed fields U and V are also distinct.