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 $\varphi :K\to \mathbb{R}$ or a pair of complexconjugate embeddings $(\psi ,\overline{\psi})$, with $\overline{\psi}\ne \psi $ and $\psi :K\to \u2102$. The archimedean places are sometimes called the infinite places (cf. place of field).

2.
The nonarchimedean places of $K$ are the prime ideals^{} in ${\mathcal{O}}_{K}$, the ring of integers^{} of $K$ (see nonarchimedean valuation (http://planetmath.org/Valuation^{})). The nonarchimedean places are sometimes called the finite places.
Notice that any archimedean place $\varphi :K\to \u2102$ can be extended to an embedding $\widehat{\varphi}:\overline{\mathbb{Q}}\to \u2102$, where $\overline{\mathbb{Q}}$ is a fixed algebraic closure^{} of $\mathbb{Q}$ (in order to prove this, one uses the fact that $\u2102$ 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 $\widehat{\varphi}:F\to \u2102$ of $F$.
Next, we define the decomposition and inertia group associated to archimedean places. For the case of nonarchimedean 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:
$${\varphi}_{1}(F)={\varphi}_{2}(F)\subset \u2102.$$ 
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:
$$D(wv)=T(wv)=\{\sigma \in \mathrm{Gal}(F/K):w\circ \sigma =w\}.$$ 
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 nonarchimedean 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 $w=\varphi :F\to \mathbb{R}$ is a real embedding. Then $\varphi $ is injective and $\varphi \circ \sigma =\varphi $ implies that $\sigma $ is the identity^{} automorphism and $T(wv)$ 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  20130322 15:07:19 
Last modified on  20130322 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  nonarchimedean place 