Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

up vote 10 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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.