The following line confused me while reading. Let $K$ be a number field. Let $\sigma_1, \ldots, \sigma_r$ denote the embeddings of $K$ in $\mathbb{R}$ and let $\tau_1, \overline{\tau}_1, \ldots, \tau_s, \overline{\tau}_s$ denote the remaining embeddings of $K$ in $\mathbb{C}.$ Thus, $r + 2s = n.$
My confusion is the following. If $K$ is normal over $\mathbb{Q}$ then can there exist both real and non-real embeddings? I don't think this is possible as the existence of a real embedding implies that there cannot be non-real roots of the minimal polynomial of $\alpha$ where $K = \mathbb{Q}[\alpha].$ So I guess if $K$ is not normal, there can be both real and non-real embeddings, but if $K$ is normal, $r$ or $s$ is 0. Is my reasoning correct?
$\Bbb Q(\sqrt[3]2)$ has both real and complex embeddings. Of course, it is not normal. A normal field extension over $\Bbb Q$ is however either totally real or totally imaginary. If it has an embedding into $\Bbb R$, all its embeddings lie inside $\Bbb R$, as you say.