Let and let be the

th cyclotomic field.

Show that is a normal extension and its Galois group is cyclic of order

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- Apr 13th 2009, 04:12 AMZetaXCyclotomic field
Let and let be the

th cyclotomic field.

Show that is a normal extension and its Galois group is cyclic of order - Apr 13th 2009, 09:50 AMThePerfectHacker
We know that .

Define, by i.e. is complex conjugation.

Let and .

Therefore, by Galois theory .

Now an element is fixed by if and only if (because ).

Therefore, .

Thus, .

This is a normal extension just notice that is an abelian group and so all its subgroups are normal.

To see that this subgroup is cyclic you need to understand group structure of .

The group is isomorphic to .

It should be clear now the Galois extension is cyclic. - Apr 13th 2009, 02:35 PMZetaX
How do you know that G is isomorphic to ?

- Apr 14th 2009, 10:22 AMThePerfectHacker