Fixed field of an automorphism group

Let be a field extension and be a group of field extension automorphisms. The fixed field of is the intermediate field #m/def/field

The induced correspondence from intermediate fields to subgroups of is called the Galois correspondence, which is an example of a Galois connection in that it is order-reversing:

Moreover, for any and ,

Further still, if and ,

where and are the generated fields and groups respectively.

Proof

The only of these which is not immediate are the last ones. Suppose . Then must also fix anything which can be written as a product of elements in ands , hence , and therefore . Since , the other inclusion is immediate.

Similarly, suppose . Then must be fixed by any product of elements from and , hence . Since , the other inclusion is immediate.

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