Galois Group Correspondence On Extension Fields Over Q

Let  be an extension field where  denotes dimension of  as a vector space over . Let  be the group of all automorphism of  that fixes  where the order of  is denoted by . Particularly, an extension field is called a Galois extension if . Moreover, we will give some properties of an extension field  which is a Galois extension. Using the properties of Galois extension, we will show that there is an one-one correspondence between the set of all intermediate fields in  and the set of all subgroups in . Furthermore, we will give some examples of Galois group correspondence using an extension field over .