PlanetMath: ramification of archimedean places

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ramification of archimedean places

(Definition)

Throughout this entry, if $\alpha$ is a complex number, we denote the complex conjugate of $\alpha$ by $\overline{\alpha}$ .

Definition 1 Let be a number field.

An archimedean place of is either a real embedding $\phi\colon K \to \mathbb{R}$ or a pair of complex-conjugate embeddings $(\psi,\overline{\psi})$ , with $\overline{\psi}\neq \psi$ and $\psi\colon K\to \mathbb{C}$ . The archimedean places are sometimes called the infinite places (cf. place of field).
The non-archimedean places of are the prime ideals in $\mathcal{O}_K$ , the ring of integers of (see non-archimedean valuation). The non-archimedean places are sometimes called the finite places.

Notice that any archimedean place $\phi\colon K\to \mathbb{C}$ can be extended to an embedding $\hat{\phi}\colon \overline{\mathbb{Q}} \to \mathbb{C}$ , where $\overline{\mathbb{Q}}$ is a fixed algebraic closure of $\mathbb{Q}$ (in order to prove this, one uses the fact that $\mathbb{C}$ is algebraically closed and also Zorn's Lemma). See also this entry. In particular, if is a finite extension of then $\phi$ can be extended to an archimidean place $\hat{\phi}\colon F \to \mathbb{C}$ of .

Next, we define the decomposition and inertia group associated to archimedean places. For the case of non-archimedean places (i.e. prime ideals) see the entries decomposition group and ramification.

Let be a finite Galois extension of number fields and let $\phi$ be a (real or a pair of complex) archimedean place of . Let $\phi_1$ and $\phi_2$ be two archimedean places of which extend $\phi$ . Notice that, since is Galois, the image of $\phi_1$ and $\phi_2$ are equal, in other words:

$\displaystyle \phi_1(F)=\phi_2(F)\subset \mathbb{C}.$

Hence, the composition $\phi_1^{-1}\circ \phi_2$ is an automorphism of

(here $\phi_1^{-1}$ denotes the inverse map of $\phi_1$ , restricted to $\phi_1(F)$ ). Thus, $\phi_1^{-1}\circ \phi_2 =\sigma \in \operatorname{Gal}(F/K)$ and

$\displaystyle \phi_2 = \phi_1\circ \sigma$

so $\phi_1$ and $\phi_2$ differ by an element of the Galois group. Similarly, if $(\psi_1,\overline{\psi_1})$ and $(\psi_2,\overline{\psi_2})$ are complex embeddings which extend $\phi$ , then there is $\sigma\in\operatorname{Gal}(F/K)$ such that

$\displaystyle (\psi_2,\overline{\psi_2})=(\psi_1,\overline{\psi_1})\circ \sigma$

meaning that either $\psi_2 = \psi_1\circ \sigma$ (and thus $\overline{\psi_2} = \overline{\psi_1}\circ \sigma$ ) or $\overline{\psi_2} = \psi_1\circ \sigma$ (and thus $\psi_2 = \overline{\psi_1}\circ \sigma$ ). We are ready now to make the definitions.

Definition 2 Let be a Galois extension of number fields and let be an archimedean place of lying above a place of . The decomposition and inertia subgroups for the pair $w\vert v$ are equal and are defined by:

$\displaystyle D(w\vert v)=T(w\vert v)=\{ \sigma \in \operatorname{Gal}(F/K) : w\circ \sigma = w\}.$

Let $e=e(w\vert v)=\vert T(w\vert v)\vert$ be the size of the inertia subgroup. If then we say that the archimedean place is ramified in the extension .

The ramification in the archimedean case is much simpler than the non-archimedean analogue. One readily proves the following proposition:

Proposition 1 The inertia subgroup $T(w\vert v)$ is nontrivial only when is real, $w=(\psi,\overline{\psi})$ is a complex archimedean place of and $\sigma$ is the “complex conjugation” map which has order . Therefore $e(w\vert v)=1$ or and ramification of archimedean places occurs if and only if there is a complex place of lying above a real place of .

Proof. Suppose first that $w=\phi\colon F\to \mathbb{R}$ is a real embedding. Then $\phi$ is injective and $\phi\circ \sigma = \phi$ implies that $\sigma$ is the identity automorphism and $T(w\vert v)$ would be trivial. So let us assume that $w=(\psi,\overline{\psi})$ is a complex archimedean place and let $\sigma\in \operatorname{Gal}(F/K)$ such that

$\displaystyle (\psi,\overline{\psi})=(\psi,\overline{\psi})\circ \sigma.$

Therefore, either $\psi=\psi\circ \sigma$ (which implies that $\sigma$ is the identity by the injectivity of $\psi$ ) or $\psi=\overline{\psi}\circ \sigma$ . The latter implies that $\sigma=\overline{\psi^{-1}}\circ \psi$ , which is simply complex conjugation:

$\displaystyle \overline{\psi^{-1}}\circ \psi(k)=\overline{\psi^{-1}(\psi(k))}=\overline{k}.$

Finally, since

is an extension of

, the equation $w\circ \sigma = w$ restricts to $\overline{v}=v$ , thus

must be real. $\qedsymbol$

Corollary 1 Suppose is an extension of number fields and assume that is a totally imaginary number field. Then the extension is unramified at all archimedean places.

Proof. Since

is totally imaginary none of the embeddings of

are real. By the proposition, only real places can ramify. $\qedsymbol$

"ramification of archimedean places" is owned by alozano.

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Other names:

finite place, infinite place

Also defines:

decomposition and inertia group for archimedean places, archimedean place, non-archimedean place

This object's parent.

Attachments:: examples of ramification of archimedean places (Example) by alozano

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Cross-references: imaginary, equation, complex conjugation, identity, implies, proposition, non-archimedean, archimedean, extension, size, subgroups, definitions, complex embeddings, Galois group, restricted, map, inverse, automorphism, composition, image, complex, real, Galois extension, ramification, decomposition group, inertia group, decomposition, finite extension, Zorn's lemma, algebraically closed, order, algebraic closure, fixed, ring of integers, prime ideals, place of field, embeddings, real embedding, number field, complex conjugate, complex number
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This is version 6 of ramification of archimedean places, born on 2005-03-09, modified 2005-03-10.
Object id is 6861, canonical name is RamificationOfArchimedeanPlaces.
Accessed 4669 times total.

Classification:

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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