I have the following exercise :
Let $p_1,...,p_n$ be distinct primes.Consider $L= \mathbb{Q}(\sqrt{p_1},...,\sqrt{p_n})$.
Find all the intermediate subfields $K$ such that $[K:\mathbb{Q}]=2$
I have already proved that $[L:\mathbb{Q}]= 2^n$
I know that if $K$ is an intermediate subfield such that $[K:\mathbb{Q}] =2$ then $[L:K] = 2^{n-1}$.
I also know that $\sqrt{p_i}\notin \mathbb{Q}(\sqrt{p_1},...,\sqrt{p_{i-1}})$ so :
$[\mathbb{Q}(\sqrt{p_1},...,\sqrt{p_{i-1}},\sqrt{p_i}): \mathbb{Q}(\sqrt{p_1},...,\sqrt{p_{i-1}})]=2 $.
But I am not seeing how to find the ones that have degree 2 over $\mathbb{Q}$