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Let $K$ be a number field, for every positive integer n, does there always exists a Galois extension $L/K$ s.t $[L:K]=n$ ?

For $K=\mathbb Q$ this is trivial, but I have no idea for other number field, does there eixsts something like Galois inverse problem that shows this is true ?

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  • $\begingroup$ There aren't always field extensions of any given integer. i.e, math.stackexchange.com/questions/615604/… $\endgroup$ – Leon Sot Dec 8 '16 at 7:03
  • $\begingroup$ @Leon Sot , this is about a real closed field $F$. One can show $F(\sqrt{-1})$ is the algebraic closure and the extension degree is 2. But a number field $K$ is a finite extension of rational numbers , so never real closed. $\endgroup$ – sawdada Dec 8 '16 at 7:16
  • $\begingroup$ Ah, my mistake, I didn't read the number field part. $\endgroup$ – Leon Sot Dec 8 '16 at 7:17
  • $\begingroup$ I don't know that it's "trivial" for $K = \mathbb{Q}$; it's straightforward if you don't require that the extension is Galois, but with that requirement the shortest argument I can think of involves subextensions of cyclotomic extensions and takes a little work. Speaking of which, maybe class field theory can answer this question. $\endgroup$ – Qiaochu Yuan Dec 8 '16 at 8:18

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