# 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:

$$\psi :K\to \u2102,\sqrt{-d}\to \sqrt{-d}$$ |

$$\overline{\psi}:K\to \u2102,\sqrt{-d}\to -\sqrt{-d}$$ |

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

$$\varphi :\mathbb{Q}\to \mathbb{R}$$ |

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 $w|v$.

Title | examples of ramification of archimedean places |
---|---|

Canonical name | ExamplesOfRamificationOfArchimedeanPlaces |

Date of creation | 2013-03-22 15:07:29 |

Last modified on | 2013-03-22 15:07:29 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 4 |

Author | alozano (2414) |

Entry type | Example |

Classification | msc 11S15 |

Classification | msc 13B02 |

Classification | msc 12F99 |

Related topic | TotallyRealAndImaginaryFields |

Related topic | ExamplesOfPrimeIdealDecompositionInNumberFields |