Can you give me a reference* for the following theorem:
Let $\mathbb{Q} \subset K\subset \mathbb{C}$ be an algebraic number field and $\alpha \in K$. If $\sigma(\alpha) = \alpha$ for all $\sigma \in \mathrm{Hom}(K, \mathbb{C})$, then $\alpha \in \mathbb{Q}$.
And:
The number of embeddings of $\mathbb{Q}(\alpha)$ in $\mathbb{C}$ equals the degree of $\alpha$.
Thank you!
*this means a book where I can find these theorems.
An embedding of $K$ in $\mathbf C$ maps $\alpha$ onto a root of the minimal polynomial of $\alpha$, and conversely, mapping $\alpha$ onto a root of the minimal polynomial of $\alpha$ defines a $\mathbf Q$-homomorphism of $\mathbf Q(\alpha)$ into $\mathbf C$ (map $\mathbf Q[x]$ into $\mathbf C$ first and check the kernel is generated by the minimal polynomial of $\alpha$).
Hence if $\sigma(\alpha)=\alpha$ for all $\sigma\in \operatorname{Hom}(K,\mathbf C)$, the only root of the minimal polynomial is $\alpha$, and it is a simple root, since we're in characteristic $0$, which proves the minimal polynomial has degree $1$ — in other words, $\alpha \in\mathbf Q$.This also proves the second assertion.