# Functoriality of the Artin Map

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 ${L}$ be the splitting field of ${x^3-x-1}$ over ${\mathbb{Q}}$. We get a non-abelian Galois group ${H\simeq S_3}$ (to keep notation the same, we called this ${H}$ last time). Take the quadratic subextension ${K=\mathbb{Q}(\sqrt{-23})}$. We have an abelian Galois group ${G\simeq \mathbb{Z}/3}$. We need the abelianization ${H^{ab}\simeq \mathbb{Z}/2}$.

By Galois theory we know ${H^{ab}}$ gives us a field extension ${L'}$ sitting between ${L}$ and ${\mathbb{Q}}$. Class field theory tells us that the conductor ${\frak{m}_{L'}=\{(23), w_1\}}$ because we must pick up all ramification from ${\mathbb{Q}(\sqrt{-23})/\mathbb{Q}}$. The general argument we gave a few posts ago shows us that ${Cl_{\frak{m}_{L'}}(\mathbb{Q})\simeq (\mathbb{Z}/23)^*}$. Now ${K}$ is the Hilbert class field, so ${Cl(\mathcal{O}_K)\stackrel{\sim}{\rightarrow} G}$ via the Artin map. We take ${\frak{m}=(\sqrt{-23})\mathcal{O}_K}$ with no embeddings specified.

This gives us the diagram:

$\displaystyle \begin{matrix} Cl_\frak{m}(K) & \twoheadrightarrow & G & \stackrel{\Psi_{L/K}}{\rightarrow} & \mathbb{Z}/3 & \rightarrow & 1 \\ id \downarrow & & & & \downarrow & & \\ Cl_\frak{m}(K) & \stackrel{N_{K/F}}{\rightarrow} & (\mathbb{Z}/23)^* & \stackrel{\Psi_{L'/F}}{\rightarrow} & \mathbb{Z}/2 & \rightarrow & 1 \end{matrix}$

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 ${(\sqrt{-23})\mathcal{O}_K}$ and taking the norm of it which lands you in the units ${(\mathbb{Z}/23)^*\simeq \mathbb{Z}/22}$. Moreover, the map ${\mathbb{Z}/22\rightarrow \mathbb{Z}/2}$ 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 ${N_{K/F}(\frak{b})}$ for any ${\frak{b}}$ prime to ${(\sqrt{-23})\mathcal{O}_K}$ is a square mod 23.

Let’s wrap up today by stating another functoriality result. Given the same setup of ${L/K/F}$ where ${G=Gal(L/K)}$ is abelian and ${H=Gal(L/F)}$ is finite possibly non-abelian. Suppose now that ${G\triangleleft H}$ and ${K/F}$ Galois with ${T=Gal(K/F)}$. Now ${T}$ acts on ${G}$ as follows. Let ${t\in T}$. Choose a lift ${h_t\in H}$. The action is given by ${t\cdot g=h_t g h_t^{-1}}$. Call this action ${\sigma_t}$.

We can transfer this Galois action to the ray class group as follows:

$\displaystyle \begin{matrix} Cl_\frak{m}(K) & \stackrel{\Psi}{\rightarrow} & G & \rightarrow & 1 \\ \downarrow & & \downarrow \sigma_t & & \\ Cl_\frak{m}(K) & \stackrel{\Psi}{\rightarrow} & G & \rightarrow & 1 \end{matrix}$

where the vertical arrow is just the natural map on ideals. Commutativity of the diagram just comes from the standard fact that ${hFrob_p(Q)h^{-1}=Frob_p(hQ)}$. 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.