Fixed fields and Galois subgroups.

1. May 9, 2013

Artusartos

1. The problem statement, all variables and given/known data

I need to find all the intermediate fields between $Q(\zeta_7)$ and Q.

2. Relevant equations

3. The attempt at a solution

For the group $Q(\zeta_7)$. I found all the permutations that are possible for $\zeta_7$.

Here is what I have:

Since $Aut(<\zeta_7>) = Z^{\times}_7 = \{1, 2, 3, 4, 5, 7\}$, we know that we can have 6 automorphisms.

Let $$\zeta = \zeta_7$$

$$\zeta \rightarrow \zeta$$

$$\zeta \rightarrow \zeta^2$$

$$\zeta \rightarrow \zeta^3$$

$$\zeta \rightarrow \zeta^4$$

$$\zeta \rightarrow \zeta^5$$

$$\zeta \rightarrow \zeta^6$$

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...

$$\zeta \rightarrow \zeta$$ has order 1

$$\zeta \rightarrow \zeta^2 \rightarrow \zeta^4 \rightarrow \zeta$$ Order=3

$$\zeta \rightarrow \zeta^3 \rightarrow \zeta^2 \rightarrow \zeta^^6 \rightarrow^4 \rightarrow \zeta^5 \rightarrow \zeta$$ Order = 6

$$\zeta \rightarrow \zeta^4 \rightarrow \zeta^2 \rightarrow \zeta$$ Order = 3

$$\zeta \rightarrow \zeta^5 \rightarrow \zeta^4 \rightarrow \zeta^6 \rightarrow \zeta^2 \rightarrow \zeta^3 \rightarrow \zeta$$ Order = 6

$$\zeta \rightarrow \zeta^6 \rightarrow \zeta$$ Order = 2

So now I need to find the fixed subgroups.

I know that $< \zeta>$ will correspond to $Q(\zeta)$ since it fixes everything.

I know that the other fields I need to look at are $Q(\zeta^2)$, $Q(\zeta^3)$, $Q(\zeta^4)$, $Q(\zeta^5)$, and $Q(\zeta^6)$.

There are two reasons why I was stuck:

1) Since, for an intermediate field K and Galois subgorup H of G, $[ K : Q] = [G : H]$, 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 $\zeta^k$ 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 $\zeta^3$, I tried to see which automorphism would give me $\zeta^3$ 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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