real and complex embeddings
Let $L$ be a subfield^{} of $\u2102$.
Definition 1.

1.
A real embedding of $L$ is an injective field homomorphism
$$\sigma :L\hookrightarrow \mathbb{R}$$ 
2.
A (nonreal) complex embedding of $L$ is an injective field homomorphism
$$\tau :L\hookrightarrow \u2102$$ such that $\tau (L)\u2288\mathbb{R}$.

3.
We denote ${\mathrm{\Sigma}}_{L}$ the set of all embeddings, real and complex, of $L$ in $\u2102$ (note that all of them must fix $\mathbb{Q}$, since they are field homomorphisms).
Note that if $\sigma $ is a real embedding then $\overline{\sigma}=\sigma $, where $\overline{\cdot}$ denotes the complex conjugation automorphism^{}:
$$\overline{\cdot}:\u2102\to \u2102,\overline{(a+bi)}=abi$$ 
On the other hand, if $\tau $ is a complex embedding, then $\overline{\tau}$ is another complex embedding, so the complex embeddings always come in pairs $\{\tau ,\overline{\tau}\}$.
Let $K\subseteq L$ be another subfield of $\u2102$. Moreover, assume that $[L:K]$ is finite (this is the dimension^{} of $L$ as a vector space^{} over $K$). We are interested in the embeddings of $L$ that fix $K$ pointwise, i.e. embeddings $\psi :L\hookrightarrow \u2102$ such that
$$\psi (k)=k,\forall k\in K$$ 
Theorem 1.
For any embedding $\psi $ of $K$ in $\mathrm{C}$, there are exactly $\mathrm{[}L\mathrm{:}K\mathrm{]}$ embeddings of $L$ such that they extend $\psi $. In other words, if $\phi $ is one of them, then
$$\phi (k)=\psi (k),\forall k\in K$$ 
Thus, by taking $\psi \mathrm{=}{\mathrm{Id}}_{K}$, there are exactly $\mathrm{[}L\mathrm{:}K\mathrm{]}$ embeddings of $L$ which fix $K$ pointwise.
Hence, by the theorem, we know that the order of ${\mathrm{\Sigma}}_{L}$ is $[L:\mathbb{Q}]$. The number $[L:\mathbb{Q}]$ is usually decomposed as
$$[L:\mathbb{Q}]={r}_{1}+2{r}_{2}$$ 
where ${r}_{1}$ is the number of embeddings which are real, and $2{r}_{2}$ is the number of embeddings which are complex (nonreal). Notice that by the remark above this number is always even, so ${r}_{2}$ is an integer.
Remark: Let $\psi $ be an embedding of $L$ in $\u2102$. Since $\psi $ is injective, we have $\psi (L)\cong L$, so we can regard $\psi $ as an automorphism of $L$. When $L/\mathbb{Q}$ is a Galois extension^{}, we can prove that ${\mathrm{\Sigma}}_{L}\cong \mathrm{Gal}(L/\mathbb{Q})$, and hence proving in a different way the fact that
$$\mid {\mathrm{\Sigma}}_{L}\mid =[L:\mathbb{Q}]=\mid \mathrm{Gal}(L/\mathbb{Q})\mid $$ 
Title  real and complex embeddings 

Canonical name  RealAndComplexEmbeddings 
Date of creation  20130322 13:54:43 
Last modified on  20130322 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 