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May 9th, 2013, 05:58 PM   #1
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Fixed fields and Galois subgroups.

For the group . I found all the permutations that are possible for .

Here is what I have:

Since , we know that we can have 6 automorphisms.

Let













I computed the orders of these automorphisms (since I want to know the subgroup orders in the Galois group in order to determine the fixed fields).

So...

has order 1

Order=3

Order = 6


Order = 3

Order = 6

Order = 2

So now I need to find the fixed subgroups.

I know that will correspond to since it fixes everything.

I know that the other fields I need to look at are , , , , and .

There are two reasons why I was stuck:

1) Since, for an intermediate field K and Galois subgorup H of G, , finding [K:Q] would help reduce the possibilities, right? But I'm kind of confused about how we would be able to find that.

2) Even if we did find [K:Q] for each K, that would only reduce the possibilites and not tell us exactly which subgorup corresponds to which intermediate field, right? So we have to check which subgroup fixes for some k between 1 and 6, right? But I don't really get anywhere with this method. I'm probably doing something wrong, but I'm not sure what. For example, for , I tried to see which automorphism would give me back but couldn't really find any that would. So I'm probably missing something, but I'm not sure why...

Thank you in advance
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