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 May 9th, 2013, 05:58 PM #1 Member   Joined: May 2012 Posts: 56 Thanks: 0 Fixed fields and Galois subgroups. 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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