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**laipou****Member**- Registered: 2009-09-19
- Posts: 24

Let

(primitive 11th root of unity).Let (primitive 5th root of unity).

Determine all intermediaye fields between and.

I have shown that actually

,((primitive 5th root of unity)),and.

So after a little calcualtion,we have ,which is isomorphic to .

Hence,the subgroups of are and.

Now comes the main problem,how to find the fixed fields of these two subgroups?

Thanks for any help.

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**Ricky****Moderator**- Registered: 2005-12-04
- Posts: 3,791

For any subgroup H, the corresponding subfield is all the elements which are fixed by any permutation in H. Hence if you take H = G you get the base field (assuming the extension is Galois), and if you take H = 1, you get the entire field extension. These are just the trivial cases.

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**laipou****Member**- Registered: 2009-09-19
- Posts: 24

Is it possible to write down the fixed field explictily(something like

)?Offline

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