I came across the following assertion in the Wikipedia article about totally imaginary number fields.
Let $K/\mathbb{Q}$ be an algebraic number field that is Galois over $\mathbb{Q}$. Then $K$ is totally real or it is totally imaginary.
Now, since I don't have any references for this result I was trying to prove it myself, and my supposed proof is the following.
What I think is that I basically only need the fact that $K/\mathbb{Q}$ is a normal extension, and then I use the following condition, that is equivalent to normality. Since $K/\mathbb{Q}$ is normal then every embedding $\sigma: K \hookrightarrow \overline{\mathbb{Q}}$ is an automorphism of $K$, which means in particular that $\sigma(K) = K$. Then since we either have $K \subset \mathbb{R}$ or else $K \cap (\mathbb{C} \setminus \mathbb{R} ) \neq \emptyset$ then this shows that $K$ is either totally real or totally imaginary.
Now my questions are if my proof is correct or if I'm missing something and if someone can provide me some references where totally real, totally imaginary and CM-fields are treated at least is some detail or where I can find some basic properties of these types of number fields.
Thank you very much.