Let $K, L$ be subfields of $\mathbb{C}$ with $K \subset L$.

Suppose that $K$ is contained in $\mathbb{R}$ and $L / K$ is a finite Galois extension of odd degree. Prove that $L$ is also contained in $\mathbb{R}$.

Give one concrete example of $K, L$ as above with $K \neq L$. Also give an example in which $K$ is contained in $\mathbb{R}$ and $L / K$ has odd degree, but is not Galois and $L$ is not contained in $\mathbb{R}$.

[Standard facts on fields and their extensions can be quoted without proof, as long as they are clearly stated.]