Throughout this entry, if $\alpha$ is a complex number, we denote the complex conjugate of $\alpha$ by $\overline{\alpha}$ .

Notice that any archimedean place $\phi\colon K\to \Complex$ can be extended to an embedding $\hat{\phi}\colon \overline{\Rats} \to \Complex$ , where $\overline{\Rats}$ is a fixed algebraic closure of $\Rats$ (in order to prove this, one uses the fact that $\Complex$ is algebraically closed and also Zorn's Lemma). See also this entry. In particular, if $F$ is a finite extension of $K$ then $\phi$ can be extended to an archimidean place $\hat{\phi}\colon F \to \Complex$ of $F$ .

Next, we define the decomposition and inertia group associated to archimedean places. For the case of non-archimedean places (i.e. prime ideals) see the entries decomposition group and ramification.

Let $F/K$ be a finite Galois extension of number fields and let $\phi$ be a (real or a pair of complex) archimedean place of $K$ . Let $\phi_1$ and $\phi_2$ be two archimedean places of $F$ which extend $\phi$ . Notice that, since $F/K$ is Galois, the image of $\phi_1$ and $\phi_2$ are equal, in other words: $$\phi_1(F)=\phi_2(F)\subset \Complex.$$ Hence, the composition $\phi_1^{-1}\circ \phi_2$ is an automorphism of $F$ (here $\phi_1^{-1}$ denotes the inverse map of $\phi_1$ , restricted to $\phi_1(F)$ ). Thus, $\phi_1^{-1}\circ \phi_2 =\sigma \in \Gal(F/K)$ and $$\phi_2 = \phi_1\circ \sigma$$ so $\phi_1$ and $\phi_2$ differ by an element of the Galois group. Similarly, if $(\psi_1,\overline{\psi_1})$ and $(\psi_2,\overline{\psi_2})$ are complex embeddings which extend $\phi$ , then there is $\sigma\in\Gal(F/K)$ such that $$(\psi_2,\overline{\psi_2})=(\psi_1,\overline{\psi_1})\circ \sigma$$ meaning that either $\psi_2 = \psi_1\circ \sigma$ (and thus $\overline{\psi_2} = \overline{\psi_1}\circ \sigma$ ) or $\overline{\psi_2} = \psi_1\circ \sigma$ (and thus $\psi_2 = \overline{\psi_1}\circ \sigma$ ). We are ready now to make the definitions.

The ramification in the archimedean case is much simpler than the non-archimedean analogue. One readily proves the following proposition:

Proof. Suppose first that $w=\phi\colon F\to \Reals$ is a real embedding. Then $\phi$ is injective and $\phi\circ \sigma = \phi$ implies that $\sigma$ is the identity automorphism and $T(w|v)$ would be trivial. So let us assume that $w=(\psi,\overline{\psi})$ is a complex archimedean place and let $\sigma\in \Gal(F/K)$ such that $$(\psi,\overline{\psi})=(\psi,\overline{\psi})\circ \sigma.$$ Therefore, either $\psi=\psi\circ \sigma$ (which implies that $\sigma$ is the identity by the injectivity of $\psi$ ) or $\psi=\overline{\psi}\circ \sigma$ . The latter implies that $\sigma=\overline{\psi^{-1}}\circ \psi$ , which is simply complex conjugation: $$\overline{\psi^{-1}}\circ \psi(k)=\overline{\psi^{-1}(\psi(k))}=\overline{k}.$$ Finally, since $w$ is an extension of $v$ , the equation $w\circ \sigma = w$ restricts to $\overline{v}=v$ , thus $v$ must be real.

Proof. Since $K$ is totally imaginary none of the embeddings of $K$ are real. By the proposition, only real places can ramify.