real and complex embeddings

Let L be a subfieldMathworldPlanetmath of .

Definition 1.
  1. 1.

    A real embedding of L is an injective field homomorphism

  2. 2.

    A (non-real) complex embedding of L is an injective field homomorphism


    such that τ(L).

  3. 3.

    We denote ΣL the set of all embeddings, real and complex, of L in (note that all of them must fix , since they are field homomorphisms).

Note that if σ is a real embedding then σ¯=σ, where ¯ denotes the complex conjugation automorphismPlanetmathPlanetmathPlanetmathPlanetmath:


On the other hand, if τ is a complex embedding, then τ¯ is another complex embedding, so the complex embeddings always come in pairs {τ,τ¯}.

Let KL be another subfield of . Moreover, assume that [L:K] is finite (this is the dimensionPlanetmathPlanetmath of L as a vector spaceMathworldPlanetmath over K). We are interested in the embeddings of L that fix K pointwise, i.e. embeddings ψ:L such that

Theorem 1.

For any embedding ψ of K in C, there are exactly [L:K] embeddings of L such that they extend ψ. In other words, if φ is one of them, then


Thus, by taking ψ=IdK, there are exactly [L:K] embeddings of L which fix K pointwise.

Hence, by the theorem, we know that the order of ΣL is [L:]. The number [L:] is usually decomposed as


where r1 is the number of embeddings which are real, and 2r2 is the number of embeddings which are complex (non-real). Notice that by the remark above this number is always even, so r2 is an integer.

Remark: Let ψ be an embedding of L in . Since ψ is injective, we have ψ(L)L, so we can regard ψ as an automorphism of L. When L/ is a Galois extensionMathworldPlanetmath, we can prove that ΣLGal(L/), and hence proving in a different way the fact that

Title real and complex embeddings
Canonical name RealAndComplexEmbeddings
Date of creation 2013-03-22 13:54:43
Last modified on 2013-03-22 13:54:43
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Definition
Classification msc 12D99
Related topic GaloisGroup
Related topic TotallyRealAndImaginaryFields
Related topic RamificationOfArchimedeanPlaces
Defines real embedding
Defines complex embedding