# Galois extension of an imaginary quadratic field

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

Question: Let $K$ be an imaginary quadratic field, and let $K\subset L$ be a Galois extension. As usual, $\tau$ will denote a complex conjugation. Then if $L$ is Galois over $Q$, then prove that

i) $[L\cap R:Q]=[L:K]$

ii)For $\alpha\in L\cap R$, $L\cap R=Q(\alpha) \Leftrightarrow L=K(\alpha)$

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 $L=M(\sqrt{-d})$.