November 8, 2011 1 Comment
Last time we started with a subgroup of the automorphism group of a field , and asked about it’s fixed field, the field of elements which are fixed by every automorphism in the subgroup. We noticed that if we took a big group, we would get a small fixed field. If we picked a small group, we would get a big fixed field.
Today I want to do the same thing but in the other direction. Instead of starting with a subgroup and finding it’s fixed field, I want to start with a field and compute the group that fixes it. Of course, we always need to do this relative to some base field, so really what we’re asking about is the is a field extension of , what are the automorphisms of which fix . We think of as being set in stone, and as being allowed to vary.
This collection of automorphisms is called the Galois Group of the extension (this is notation for saying that is an extension of ). We denote it by . More techncially,
Of course, we need to really check that this is a group, but it’s not so bad. If two automorphisms both fix , then so does their composition (first the first one fixes , then the second fixes ). It’s similarly easy to check for inverses and identity. This proof is neither difficult nor enlightening. I’d rather get to the math that is both.
Let’s work out a simple example. Let be a field. What is ? These are the automorphism of which fix . There’s exactly one of those. The identity (or lame) automorphism.
There’s something else you may notice. is a particularly small extension. However, I’m thinking of it as where . Furthermore, I’m thinking of being set in stone. So then the subfield is particularly big inside . This is the correct notion, because, I really do want to think of the bigger field as unchanging. So a very big subfield (as big as they come) gives us a very small Galois group (as small as they come). This is true in general too. Small subfields give big Galois groups and big subfields give small Galois groups.
This idea of matching groups with fields in such a way that big groups get matched with small fields and vice versa is incredibly important and incredibly powerful. We can now start exploring just how deep the rabbit hole goes.