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

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up vote 11 down vote accepted

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.

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