Embedding of a number field (reviewed)

An embedding of a number field $K$ is a field homomorphism $K\to \C$. A number field of degree $n$ has $n$ distinct embeddings, which may be distinguished as real or complex depending on whether the image of the embedding is contained in $\R$ or not.

Complex embeddings necessarily come in conjugate pairs. The signature of a number field is determined by the number of real embeddings and the number of pairs of conjugate complex embeddings.

For $K=\Q(a)$, where $a$ is an algebraic number with minimal polynomial $f(X)$, each embedding $\iota$ is uniquely determined by the value $z=\iota(a)$, which is one of the complex roots of $f(X)$. The embedding is real if $z\in\R$ and complex if $z\notin\R$.