I want to construct degree $n$ field extension with no intermediate field for each $n$. I know for any finite group $G$ there is a Galois extension $K/F$ so that $Gal(K/F)$ is $G$. So my idea was to show that for every $n$ there is a simple group (say $A_{k}$) which contains a subgroup of index $n$. But I can't proceed further. Any help?

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    $\begingroup$ This has been answered on the site. Construct an extension $K/F$ with Galois group $S_n$. Let $L$ be the fixed field of $S_{n-1}$ (a point stabilizer). Then $L/F$ cannot have any intermediate fields, because $S_{n-1}$ is a maximal subgroup. In other words, if $f(x)\in F[x]$ has Galois group $S_n$, and $\alpha$ is a zero of $f$, then $F(\alpha)/F$ has degree $n$, and no intermediate fields. Not posting that as an answer, because this is a duplicate. $\endgroup$ – Jyrki Lahtonen Nov 21 '14 at 7:02
  • $\begingroup$ Adding an important detail: $f(x)$ above should be irreducible of degree $n$. It was implicitly there in the claim that $[F(\alpha):F]=n$, but it is best to make that explicit. $\endgroup$ – Jyrki Lahtonen Nov 21 '14 at 10:44

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