When $L/\mathbb{Q}$ is a finite extension of $\mathbb{Q}$ that doesn't lie in $\mathbb{R}$. Is there anything we can say about the degree of the extension $L/ L \cap \mathbb{R}$?
I know that for cyclotomic extensions $\mathbb{Q}(\zeta) \cap \mathbb{R} = \mathbb{Q}(\zeta + {\zeta}^{-1})$. So we have $[\mathbb{Q}(\zeta)/ \mathbb{Q}(\zeta) \cap \mathbb{R}]=2$.
Does it hold in general that $[L / L \cap \mathbb{R}]=2$? If not, are there any assumptions on $L$ that make it hold?
Can we say anything about the degree of the extension $L/ L \cap \mathbb{Q}(i \mathbb{R})$?
It's one of the fundamental results of Galois theory that for any field $ K $ and any finite group of automorphisms $ G $ of this field, we have $ [K : K_G] = |G| $ where $ K_G $ denotes the subfield of $ K $ fixed by the action of $ G $. Applying this to your example with $ L $ and the order $ 2 $ group of automorphisms generated by complex conjugation gives $ [L : L \cap \mathbb R] = 2 $ whenever $ L $ is not contained in $ \mathbb R $.