# The Galois side of the Norm map

Let $K$ be an abelian extension of $\mathbb{Q}$. We know that $[x, K]|_{\mathrm{Gal}{\mathbb{Q}^{ab}}}=[\mathrm{N}^{K}_{\mathbb{Q}} x, \mathbb{Q}]$ where $[x, F]$ is the Artin reciprocity map. Given a modulus $\mathfrak{m}$ of $K$, we get a modulus $\mathrm{N}^{K}_{\mathbb{Q}} \mathfrak{m}$. While I can write down abstractly how the Galois groups of the corresponding ray class fields are related, I don't know how the fields are related: will they embed in an easily understood manner?

• I'm not sure I understand your question. Take $K = {\mathbb Q}(i)$ and ${\mathfrak m} = (1+2i)$, then the ray class field of $K$ modulo ${\mathfrak m}$ is trivial, the ray class field of ${\mathbb Q}$ modulo $5$ is ${\mathbb Q}(\sqrt{5})$. What do you expect? – Franz Lemmermeyer Apr 6 '16 at 18:05