Field theory MOC

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 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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