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[parent] examples of ramification of archimedean places (Example)
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:
$\displaystyle \psi\colon K \to \mathbb{C}, \sqrt{-d} \to \sqrt{-d}$
$\displaystyle \overline{\psi}\colon K \to \mathbb{C}, \sqrt{-d} \to - \sqrt{-d}$
The archimedean place $ w=(\psi,\overline{\psi})$ is lying above the unique archimedean place of $ \mathbb{Q}$:
$\displaystyle \phi\colon \mathbb{Q}\to \mathbb{R}$
and therefore, the place $ v=\phi$ ramifies in $ K$.
Example 2   Let $ K$ be a CM-field i.e. $ K$ is a totally imaginary 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\vert v)=2$ for all $ w\vert v$.



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See Also: totally real and imaginary fields, examples of prime ideal decomposition in number fields


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Cross-references: complex embeddings, imaginary, real, archimedean, totally ramified, extension, totally real field, quadratic extension, CM-field, ramifies, place, archimedean place, embeddings, quadratic imaginary number field

This is version 1 of examples of ramification of archimedean places, born on 2005-03-09.
Object id is 6864, canonical name is ExamplesOfRamificationOfArchimedeanPlaces.
Accessed 943 times total.

Classification:
AMS MSC11S15 (Number theory :: Algebraic number theory: local and $p$-adic fields :: Ramification and extension theory)
 13B02 (Commutative rings and algebras :: Ring extensions and related topics :: Extension theory)
 12F99 (Field theory and polynomials :: Field extensions :: Miscellaneous)

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