# 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:

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

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

 $\phi\colon\mathbb{Q}\to\mathbb{R}$

and therefore, the place $v=\phi$ 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 $w|v$.

Title examples of ramification of archimedean places ExamplesOfRamificationOfArchimedeanPlaces 2013-03-22 15:07:29 2013-03-22 15:07:29 alozano (2414) alozano (2414) 4 alozano (2414) Example msc 11S15 msc 13B02 msc 12F99 TotallyRealAndImaginaryFields ExamplesOfPrimeIdealDecompositionInNumberFields