Galois extension of an imaginary quadratic field

Question

This is an exercise problem from the book Primes of the form $x^2+ny^2$: Fermat, Class field Theory and Complex multiplication.

Let $K$ be an imaginary quadratic field, and let $K\subset L$ be a Galois extension. As usual, $\tau$ will denote the complex conjugation. Then, if $L$ is Galois over $Q$ prove that
i) $[L\cap R:\mathbb Q]=[L:K]$.
ii) For $\alpha\in L\cap R$, $L\cap R=\mathbb Q(\alpha) \Longleftrightarrow L=K(\alpha)$.

Thanks in advance

Answer 1

Hints:

1. If $L$ is Galois over $\Bbb{Q}$, then $\tau\in Gal(L/\Bbb{Q})$. What does basic Galois correspondence tell you about the fixed field of $\langle\tau\rangle$?
2. Denoting $M=L\cap\Bbb{R}$. We were given that $K=\Bbb{Q}(\sqrt{-d})$ for some integer $d>0$. Show that $L=M(\sqrt{-d})$.