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[parent] examples of ramification of archimedean places (Example)
Example 1   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 $K$ .
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 $e(w|v)=2$ for all $w|v$ .




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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 1274 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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