examples of ramification of archimedean placesExamplesOfRamificationOfArchimedeanPlaces2005-03-09 17:55:362005-03-09 17:55:36Exampleramification of archimedean places% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Rats}{\mathbb{Q}}\begin{exa}
Let $K=\Rats(\sqrt{-d})$ be a quadratic imaginary number field. Then $K$ has only two embeddings which, in fact, are complex-conjugate embeddings:
$$\psi\colon K \to \Complex, \sqrt{-d} \to \sqrt{-d}$$
$$\overline{\psi}\colon K \to \Complex, \sqrt{-d} \to - \sqrt{-d}$$
The archimedean place $w=(\psi,\overline{\psi})$ is lying above the unique archimedean place of $\Rats$:
$$\phi\colon \Rats \to \Reals$$
and therefore, the place $v=\phi$ ramifies in $K$.
\end{exa}
\begin{exa}
Let $K$ be a CM-field i.e. $K$ is a \PMlinkname{totally imaginary}{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 $w|v$.
\end{exa}