PlanetMath: examples of ramification of archimedean places

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examples of ramification of archimedean places

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Example 1 Let $K=\mathbb{Q}(\sqrt{-d})$ be a quadratic imaginary number field. Then

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

Example 2 Let

be a CM-field i.e.

. Then we claim that the extension

is totally ramified at the archimedean (or infinite) places. Indeed, let

be an archimedean place of

. By assumption,

is a totally real field, thus all its places are real, and so,

is real. Let

be any archimedean place of

lying above

(i.e. extending

). Since

is totally imaginary, the place

is a pair of complex embeddings, and therefore

ramifies in

. Thus, all archimedean places of

ramify in

and $e(w\vert v)=2$ for all $w\vert v$ .

"examples of ramification of archimedean places" is owned by alozano.

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AMS MSC:	11S15 (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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