examples of ramification of archimedean places

Example 1.

Let $K=\mathbb{Q}(\sqrt{-d})$ be a quadratic imaginary number field. Then $K$ has only two embeddings which, in fact, are complex-conjugate embeddings:

$$\psi :K\to ℂ,\sqrt{-d}\to \sqrt{-d}$$

$$\overline{\psi }:K\to ℂ,\sqrt{-d}\to -\sqrt{-d}$$

The archimedean place $w=(\psi ,\overline{\psi })$ is lying above the unique archimedean place of $\mathbb{Q}$:

$$\varphi :\mathbb{Q}\to ℝ$$

and therefore, the place $v=\varphi$ ramifies in $K$.

Example 2.

Let $K$ be a CM-field i.e. $K$ is a totally imaginary (http://planetmath.org/TotallyRealAndImaginaryFields) quadratic extension of a totally real field $K^{+}$. Then we claim that the extension $K/K^{+}$ is totally ramified at the archimedean (or infinite) places. Indeed, let $v$ be an archimedean place of $K^{+}$. By assumption, $K^{+}$ is a totally real field, thus all its places are real, and so, $v$ is real. Let $w$ be any archimedean place of $K$ lying above $v$ (i.e. extending $v$ to $K$). Since $K$ is totally imaginary, the place $w$ is a pair of complex embeddings, and therefore $v$ ramifies in $K/K^{+}$. Thus, all archimedean places of $K^{+}$ ramify in $K$ and $e(w|v)=2$ for all $w|v$.