Let $m$ be a negative integer, and suppose that $K$ is a quartic extension over $\mathbb{Q}$ contains $\mathbb{Q}(\sqrt{m})$ as a subfield.
In this situation, I have two questions:
Q1) Is the existence of such an quartic extension $K$ over $\mathbb{Q}$ implies the positive integer $n:=-m$ is not a square in $\mathbb{Q}$?
Q2) If the first question is true, I wonder if $K$ is necessarily a biquadratic extension over $\mathbb{Q}$ of the form $\mathbb{Q}(\sqrt{n},i)$, and thus, $K$ is Galois over $\mathbb{Q}$ with $\textrm{Gal}(K/\mathbb{Q})\cong\mathbb{Z}_{2}\times\mathbb{Z}_{2}$.
I've been thinking a lot of time, but I'm not sure about my questions.
A little advice would help me a lot! Thank you.