Instead of restating the functoriality of the Artin map, let’s just review the statement through an example. We’ll re-use our example from last time. Let be the splitting field of over . We get a non-abelian Galois group (to keep notation the same, we called this last time). Take the quadratic subextension . We have an abelian Galois group . We need the abelianization .
By Galois theory we know gives us a field extension sitting between and . Class field theory tells us that the conductor because we must pick up all ramification from . The general argument we gave a few posts ago shows us that . Now is the Hilbert class field, so via the Artin map. We take with no embeddings specified.
This gives us the diagram:
First, the right vertical arrow is clearly the zero map. The other important part of the diagram is that the norm map is taking a fractional ideal (class) that is relatively prime to and taking the norm of it which lands you in the units . Moreover, the map is the unique surjective one and the image of the norm map must land in the kernel of this by exactness. Interestingly, this tells us that the positive generator of for any prime to is a square mod 23.
Let’s wrap up today by stating another functoriality result. Given the same setup of where is abelian and is finite possibly non-abelian. Suppose now that and Galois with . Now acts on as follows. Let . Choose a lift . The action is given by . Call this action .
We can transfer this Galois action to the ray class group as follows:
where the vertical arrow is just the natural map on ideals. Commutativity of the diagram just comes from the standard fact that . There is another functoriality we could do, but it doesn’t seem worth it at this point because it is overly complicated and there isn’t a plan to use it anytime soon.