Construct degree $n$ field extension with no intermediate field

Question

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?