examples of ramification of archimedean places

Example 1.

Let K=(-d) be a quadratic imaginary number field. Then K has only two embeddingsPlanetmathPlanetmath which, in fact, are complex-conjugate embeddings:


The archimedean place w=(ψ,ψ¯) is lying above the unique archimedean place of :


and therefore, the place v=ϕ 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 extensionPlanetmathPlanetmathPlanetmath K/K+ is totally ramified at the archimedeanPlanetmathPlanetmath (or infiniteMathworldPlanetmathPlanetmath) places. Indeed, let v be an archimedean place of K+. By assumptionPlanetmathPlanetmath, 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