Definition of an F-embedding

Question

I am reviewing field theory and while studying separable extensions, I found the following definition:

Definition. Let $F$ be a field and $K$ and $E$ extensions of $F$. Denote by $\operatorname{Emb}_F(K, E)$ the set of $F$-embeddings of $K$ into $E$.

I was wondering if anyone could explain what exactly is an embedding and how it differs from an $F$-embedding? Thanks for your help!

Answer 1

An embedding is just an injective ring-homomorphism (since the domain is a field, if and only if it is non-zero) $\phi:K\hookrightarrow E$. A $F$-embedding is an embedding $\phi:K\hookrightarrow E$ such that $\phi(x)=x$ for all $x\in F$. Equivalently, one may say that it is an injective homomophism $K\hookrightarrow E$ as $F$-algebras. In field theory it is often called an embedding over $F$.

For instance, consider these embeddings $\Bbb C(t)\to\Bbb C(t)$:

1. $\psi:\quad f(t)\mapsto f(-t)$

2. $\rho:\quad\dfrac{a_nt^n+\cdots +a_0}{b_mt^n+\cdots +b_0}\mapsto \dfrac{\overline a_nt^n+\cdots +\overline a_0}{\overline b_mt^m+\cdots +\overline b_0}$

3. $\rho\circ\psi$

These three embeddings are all $\neq\operatorname{id}$; you can easily see that $\psi$ is a $\Bbb C(t^2)$-embedding, $\rho$ is a $\Bbb R(t)$-embedding and $\rho\circ\psi$ is a $\Bbb R(t^2)$-embedding.