This is the lemma we shall use in the proof of the reciprocity law, to reduce the cyclic case to the cyclotomic case:

Lemma 4 (Artin)Let be a cyclic extension of degree and a prime of unramified in . Then we can find a field , a subextension of , with such that in the lattice of fields

we have:

1. splits completely in

2. , so that is cyclotomic

3. is unramified in

Moreover, we can choose such that it is divisible only by arbitrarily large primes.

The proof of this will use the previous number-theory lemmas and the basic tools of Galois theory.

So, first of all, we know that is a subextension of some . We don’t know what is, but pretend we do, and will start carrying out the proof. As we do so, we will learn more and more about what has to be like, and eventually choose it.

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