We shall now consider a number field {k} and an abelian extension {L}. Let {S} be a finite set of primes (nonarchimedean valuations) of {k} containing the ramified primes, and consider the group {I(S)} of fractional ideals prime to the elements of {S}. This is a free abelian group on the primes not in {S}. We shall define a map, called the Artin map from {I(S) \rightarrow G(L/k)}.

1. How does this work?

Specifically, let {\mathfrak{p} \notin S} be a prime in {k}. There is a prime {\mathfrak{P}} of {L} lying above it. If {A,B} are the rings of integers in {k,L}, respectively, then we have a field extension {A/\mathfrak{p} \rightarrow  B/\mathfrak{P}}. As is well-known, there is a surjective homomoprhism of the decomposition group {G_{\mathfrak{P}}} of {\mathfrak{P}} onto {G(B/\mathfrak{P} /  A/\mathfrak{p})} whose kernel, called the inertia group, is of degree {e(\mathfrak{P}|\mathfrak{p})}.

But, we know that the extension {B/\mathfrak{P} /  A/\mathfrak{p}} is cyclic, because these are finite fields. The Galois group is generated by a canonically determined Frobenius element which sends {a \rightarrow a^{|A/\mathfrak{p}|}}. We can lift this to an element {\sigma_{\mathfrak{p}}} of {G_{\mathfrak{P}}}, still called the Frobenius element. (more…)