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.