Totally real (reviewed)

A global number field $K$ is always of the form $K=\Q(\alpha)$ where $\alpha$ has monic irreducible polynomial $f(x)\in\Q[x]$. The field is totally real if all of the roots of $f(x)$ in $\C$ lie in the real numbers $\R$.

Equivalently, $K$ is totally real if all the embeddings of $K$ into $\C$ have image contained in $\R$.