ramification of archimedean places

Throughout this entry, if α is a complex numberMathworldPlanetmathPlanetmath, we denote the complex conjugateMathworldPlanetmath of α by α¯.

Definition 1.

Let K be a number fieldMathworldPlanetmath.

  1. 1.

    An archimedean place of K is either a real embedding ϕ:K or a pair of complex-conjugate embeddings (ψ,ψ¯), with ψ¯ψ and ψ:K. The archimedean places are sometimes called the infinite places (cf. place of field).

  2. 2.

    The non-archimedean places of K are the prime idealsPlanetmathPlanetmath in 𝒪K, the ring of integersMathworldPlanetmath of K (see non-archimedean valuation (http://planetmath.org/ValuationMathworldPlanetmath)). The non-archimedean places are sometimes called the finite places.

Notice that any archimedean place ϕ:K can be extended to an embedding ϕ^:¯, where ¯ is a fixed algebraic closureMathworldPlanetmath of (in order to prove this, one uses the fact that is algebraically closed and also Zorn’s Lemma). See also this entry (http://planetmath.org/PlaceAsExtensionOfHomomorphism). In particular, if F is a finite extensionMathworldPlanetmath of K then ϕ can be extended to an archimidean place ϕ^:F 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 extensionMathworldPlanetmath of number fields and let ϕ be a (real or a pair of complex) archimedean place of K. Let ϕ1 and ϕ2 be two archimedean places of F which extend ϕ. Notice that, since F/K is Galois, the image of ϕ1 and ϕ2 are equal, in other words:


Hence, the composition ϕ1-1ϕ2 is an automorphismPlanetmathPlanetmathPlanetmathPlanetmath of F (here ϕ1-1 denotes the inverse map of ϕ1, restricted to ϕ1(F)). Thus, ϕ1-1ϕ2=σGal(F/K) and


so ϕ1 and ϕ2 differ by an element of the Galois groupMathworldPlanetmath. Similarly, if (ψ1,ψ1¯) and (ψ2,ψ2¯) are complex embeddings which extend ϕ, then there is σGal(F/K) such that


meaning that either ψ2=ψ1σ (and thus ψ2¯=ψ1¯σ) or ψ2¯=ψ1σ (and thus ψ2=ψ1¯σ). We are ready now to make the definitions.

Definition 2.

Let F/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 subgroupsMathworldPlanetmathPlanetmath for the pair w|v are equal and are defined by:


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

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

Proposition 1.

The inertia subgroup T(w|v) is nontrivial only when v is real, w=(ψ,ψ¯) is a complex archimedean place of F and σ is the “complex conjugation” map which has order 2. Therefore e(w|v)=1 or 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.


Suppose first that w=ϕ:F is a real embedding. Then ϕ is injective and ϕσ=ϕ implies that σ is the identityPlanetmathPlanetmathPlanetmath automorphism and T(w|v) would be trivial. So let us assume that w=(ψ,ψ¯) is a complex archimedean place and let σGal(F/K) such that


Therefore, either ψ=ψσ (which implies that σ is the identity by the injectivity of ψ) or ψ=ψ¯σ. The latter implies that σ=ψ-1¯ψ, which is simply complex conjugation:


Finally, since w is an extension of v, the equation wσ=w restricts to v¯=v, thus v must be real. ∎

Corollary 1.

Suppose L/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/K is unramified at all archimedean places.


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