PlanetMath: real and complex embeddings

(more info)

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real and complex embeddings

(Definition)

Let be a subfield of $\mathbb{C}$ .

Definition 1

A real embedding of is an injective field homomorphism
$\displaystyle \sigma\colon L \hookrightarrow \mathbb{R}$
A (non-real) complex embedding of is an injective field homomorphism
$\displaystyle \tau\colon L \hookrightarrow \mathbb{C}$
such that $\tau(L)\nsubseteq \mathbb{R}$ .
We denote $\Sigma_L$ the set of all embeddings, real and complex, of in $\mathbb{C}$ (note that all of them must fix $\mathbb{Q}$ , since they are field homomorphisms).

Note that if $\sigma$ is a real embedding then $\bar{\sigma}=\sigma$ , where $\overline{\cdot}$ denotes the complex conjugation automorphism:

$\displaystyle \overline{\cdot}\colon \mathbb{C}\to \mathbb{C},\quad \overline{(a+bi)}=a-bi$

On the other hand, if $\tau$ is a complex embedding, then $\bar{\tau}$ is another complex embedding, so the complex embeddings always come in pairs $\{\tau,\bar{\tau}\}$ .

Let $K\subseteq L$ be another subfield of $\mathbb{C}$ . Moreover, assume that is finite (this is the dimension of as a vector space over ). We are interested in the embeddings of that fix pointwise, i.e. embeddings $\psi\colon L \hookrightarrow \mathbb{C}$ such that

$\displaystyle \psi(k)=k,\quad \forall k\in K$

Theorem 1 For any embedding $\psi$ of in $\mathbb{C}$ , there are exactly embeddings of such that they extend $\psi$ . In other words, if $\varphi$ is one of them, then

$\displaystyle \varphi(k)=\psi(k),\quad \forall k\in K$

Thus, by taking $\psi=\operatorname{Id}_K$ , there are exactly embeddings of which fix pointwise.

Hence, by the theorem, we know that the order of $\Sigma_L$ is $[L:\mathbb{Q}]$ . The number $[L:\mathbb{Q}]$ is usually decomposed as

$\displaystyle [L:\mathbb{Q}]=r_1+2r_2$

where

is the number of embeddings which are real, and

is the number of embeddings which are complex (non-real). Notice that by the remark above this number is always even, so

is an integer.

Remark: Let $\psi$ be an embedding of in $\mathbb{C}$ . Since $\psi$ is injective, we have $\psi(L)\cong L$ , so we can regard $\psi$ as an automorphism of . When $L/\mathbb{Q}$ is a Galois extension, we can prove that $\Sigma_L\cong \operatorname{Gal}(L/\mathbb{Q})$ , and hence proving in a different way the fact that