Let and let be the
th cyclotomic field.
Show that is a normal extension and its Galois group is cyclic of order
Define, by i.e. is complex conjugation.
Let and .
Therefore, by Galois theory .
Now an element is fixed by if and only if (because ).
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.