Let $a$ and $b$ be algebraic over $\mathbb{Q}$ with $a,b\in\mathbb{R}~(a,b\neq0)$.
Now, suppose that $K=\mathbb{Q}(a+bi)$ be a Galois extension over $\mathbb{Q}$, and that $\mathbb{Q}(a^{2}+b^{2})\subseteq K$.
Show that the order of the Galois group $G:=\textrm{Gal}(K/\mathbb{Q})$ is even.
In this problem, I have the following two questions:
Q1) Since $K$ is Galois over $\mathbb{Q}$, is it true that there is an element $\sigma\in G$ so that $\sigma(a+bi)=a-bi$?
Q2) If the question 1 is true, why do we need the condition `$\mathbb{Q}(a^{2}+b^{2})\subseteq K$' ?
Give some advice and comment. Thank you!