Embedding of a field extension to another

Question

Can $\mathbb{Q}(\sqrt {-2})$ be embedded into a cyclic extension of degree 4 over $\mathbb{Q}$?

Answer 1

No. Let $L$ be such an extension and choose an embedding $L \to \mathbb{C}$. Then complex conjugation induces an order two automorphism of $L$; the fixed field of this automorphism isn't $\mathbb{Q}(\sqrt{-2})$ and so the Galois group of $L$ over $\mathbb{Q}$ has more than one subgroup of order 2 and thus cannot be cyclic.