We shall now consider a number field and an abelian extension . Let be a finite set of primes (nonarchimedean valuations) of containing the ramified primes, and consider the group of fractional ideals prime to the elements of . This is a free abelian group on the primes not in . We shall define a map, called the **Artin map** from .

**1. How does this work? **

Specifically, let be a prime in . There is a prime of lying above it. If are the rings of integers in , respectively, then we have a field extension . As is well-known, there is a surjective homomoprhism of the decomposition group of onto whose kernel, called the **inertia group**, is of degree .

But, we know that the extension is cyclic, because these are finite fields. The Galois group is generated by a canonically determined **Frobenius element** which sends . We can lift this to an element of , still called the Frobenius element. (more…)