We continue (and finish) the proof started in the previous post of the second inequality.

**3. Construction of **

So, here’s the situation. We have a cyclic extension of degree , a prime number, and contains the -th roots of unity. In particular, we can write (by Kummer theory) for a subgroup of such that , in particular can be taken to be generated by one element .

We are going to prove that the norm index of the ideles is at most . Then, by the reductions made earlier, we will have proved the second inequality.

**3.1. Setting the stage**

Now take a huge but finite set of primes such that:

1.

2. is an -unit

3. contains all the primes dividing

4. contains the ramified primes We will now find a bigger extension of whose degree is a prime power. We consider the tower for the -units. We have the extension whose degree we can easily compute; it is