Are field embeddings unique? Sorry if this is a simple question, as I'm not well versed in field theory.
Suppose a field $K$ has an embedding into $\mathbb R$: $f:K\hookrightarrow\mathbb R$. Is $f$ unique? And if $\mathbb R$ is replaced by an arbitrary field $F$, is the answer still the same?

$f$ is not unique in general. For instance, there are two embeddings of $\mathbb Q[x]/(x^2-2)$ into $\mathbb R$.
In general, if $K=\mathbb F(\alpha)$, where $\mathbb F$ is the prime field of $K$, and $F$ is a field with the same prime field, then then number of $\mathbb F$-embeddings of $K$ into $F$ is the number of roots in $F$ of the minimal polynomial of $\alpha$.
