ramification of archimedean places

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

Definition 1.

Let $K$ be a number field.

1. An archimedean place of $K$ is either a real embedding $\varphi :K\to ℝ$ or a pair of complex-conjugate embeddings $(\psi ,\overline{\psi })$, with $\overline{\psi }\ne \psi$ and $\psi :K\to ℂ$. The archimedean places are sometimes called the infinite places (cf. place of field).
   

2. The non-archimedean places of $K$ are the prime ideals in ${𝒪}_K$, the ring of integers of $K$ (see non-archimedean valuation (http://planetmath.org/Valuation)). The non-archimedean places are sometimes called the finite places.

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

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 $F/K$ be a finite Galois extension of number fields and let $\varphi$ be a (real or a pair of complex) archimedean place of $K$. Let ${\varphi }_1$ and ${\varphi }_2$ be two archimedean places of $F$ which extend $\varphi$. Notice that, since $F/K$ is Galois, the image of ${\varphi }_1$ and ${\varphi }_2$ are equal, in other words:

$${\varphi }_1(F)={\varphi }_2(F)\subset ℂ.$$

Hence, the composition ${\varphi }_1^{-1}\circ {\varphi }_2$ is an automorphism of $F$ (here ${\varphi }_1^{-1}$ denotes the inverse map of ${\varphi }_1$, restricted to ${\varphi }_1(F)$). Thus, ${\varphi }_1^{-1}\circ {\varphi }_2=\sigma \in \mathrm{Gal}(F/K)$ and

$${\varphi }_2={\varphi }_1\circ \sigma$$

so ${\varphi }_1$ and ${\varphi }_2$ 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 $\varphi$, then there is $\sigma \in \mathrm{Gal}(F/K)$ such that

$$({\psi }_2,\overline{{\psi }_2})=({\psi }_1,\overline{{\psi }_1})\circ \sigma$$

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 $F\mathrm{/}K$ be a Galois extension of number fields and let $w$ be an archimedean place of $F$ lying above a place $v$ of $K$. The decomposition and inertia subgroups for the pair $w\mathrm{|}v$ are equal and are defined by:

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

Let $e\mathrm{=}e\mathrm{(}w\mathrm{|}v\mathrm{)}\mathrm{=}\mathrm{|}T\mathrm{(}w\mathrm{|}v\mathrm{)}\mathrm{|}$ be the size of the inertia subgroup. If $e\mathrm{>}\mathrm{1}$ then we say that the archimedean place $v$ is ramified in the extension $F\mathrm{/}K$.

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 $T\mathrm{(}w\mathrm{|}v\mathrm{)}$ is nontrivial only when $v$ is real, $w\mathrm{=}\mathrm{(}\psi \mathrm{,}\overline{\psi }\mathrm{)}$ is a complex archimedean place of $F$ and $\sigma$ is the “complex conjugation” map which has order $\mathrm{2}$. Therefore $e\mathrm{(}w\mathrm{|}v\mathrm{)}\mathrm{=}\mathrm{1}$ or $\mathrm{2}$ and ramification of archimedean places occurs if and only if there is a complex place of $F$ lying above a real place of $K$.

Proof.

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

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

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:

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

Finally, since $w$ is an extension of $v$, the equation $w\circ \sigma =w$ restricts to $\overline{v}=v$, thus $v$ must be real. ∎

Corollary 1.

Suppose $L\mathrm{/}K$ is an extension of number fields and assume that $K$ is a totally imaginary (http://planetmath.org/TotallyRealAndImaginaryFields) number field. Then the extension $L\mathrm{/}K$ is unramified at all archimedean places.

Proof.

Since $K$ is totally imaginary none of the embeddings of $K$ are real. By the proposition, only real places can ramify. ∎

Title	ramification of archimedean places
Canonical name	RamificationOfArchimedeanPlaces
Date of creation	2013-03-22 15:07:19
Last modified on	2013-03-22 15:07:19
Owner	alozano (2414)
Last modified by	alozano (2414)
Numerical id	9
Author	alozano (2414)
Entry type	Definition
Classification	msc 12F99
Classification	msc 13B02
Classification	msc 11S15
Synonym	finite place
Synonym	infinite place
Related topic	DecompositionGroup
Related topic	PlaceOfField
Related topic	RealAndComplexEmbeddings
Related topic	PlaceAsExtensionOfHomomorphism
Defines	decomposition and inertia group for archimedean places
Defines	archimedean place
Defines	non-archimedean place