November 4, 2011 2 Comments
If I have a field and a field automorphism , I can ask about the fixed points of . That is, all of the such that . Notice that if fixes and , then and , so fixes and . In fact, the set of fixed points of is itself a field! I just checked two of the rules. You can check the rest.
This field is called the fixed field of , and it’s super super important. If we have two automorphisms, and , we can take the intersection of their fixed fields. The intersection of fields is always a field. What field is it? It’s the field consisting of everything fixed by both and .
But why stop at two automorphisms? What if I take a group of automorphisms of . That has a fixed field. It’s called the fixed field of . I could take the entire automorphism group of , or I could just take a subgroup of it. The bigger the group I take, the more fields I intersect, so the smaller my fixed field is.
That’s an important idea. Big group means small fixed field. Small group means big fixed field. We’ll make this precise very soon.