Fixed fields and Galois subgroups.

[Artusartos]

Homework Statement

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

Homework Equations

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

 

[mighty2000]

for any help!

Hello! It seems like you are on the right track with finding the intermediate fields between Q(\zeta_7) and Q. I can offer some suggestions to help you find the fixed subgroups and corresponding intermediate fields.

To find the fixed subgroups, we can use the fact that they correspond to the intermediate fields. For example, the subgroup that fixes Q(\zeta) will correspond to the intermediate field Q(\zeta). We can also use the fact that the fixed subgroups are normal in the Galois group. This means that for any intermediate field K, the corresponding subgroup will be normal in the Galois group. So we can look for normal subgroups of the Galois group that contain the subgroup <\zeta>, which corresponds to Q(\zeta).

To find the intermediate fields corresponding to the fixed subgroups, we can use the fact that the degree of the intermediate field over Q is equal to the index of the corresponding subgroup in the Galois group. So for example, if we find a subgroup of index 2 in the Galois group, then the corresponding intermediate field will have degree 2 over Q.

I hope this helps you in finding the intermediate fields between Q(\zeta_7) and Q. Good luck!