Fixed fields

If I have a field k and a field automorphism \sigma, I can ask about the fixed points of \sigma. That is, all of the x\in k such that \sigma(x)=x. Notice that if \sigma fixes x and y, then \sigma(x+y)=\sigma(x)+\sigma(y)=x+y and \sigma(xy)=\sigma(x)\sigma(y)=xy, so \sigma fixes x+y and xy. In fact, the set of fixed points of \sigma is itself a field! I just checked two of the rules. You can check the rest.

This field is called the fixed field of \sigma, and it’s super super important. If we have two automorphisms, \sigma and \tau, 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 \sigma and \tau.

But why stop at two automorphisms? What if I take a group G of automorphisms of k. That has a fixed field. It’s called the fixed field of G. I could take the entire automorphism group of k, 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.

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2 Responses to Fixed fields

  1. Pingback: Galois Group « Andy Soffer

  2. Pingback: Galois Connection « Andy Soffer

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