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I am looking for a degree $4$ extension of $\mathbb {Q}$ with no intermediate field. I know such extension is not Galois (equivalently not normal). So I was trying to adjoin a root of an irreducible quartic. But I got stuck. Any hint/idea/solution?

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With regard to Sebastian Schoennenbeck's comment, an extension of $\mathbb{Q}$ with Galois group $A_4$ (alternating group on 4 points) will do the trick. Such an extension certainly exists, in fact all alternating groups are Galois groups over $\mathbb{Q}$.

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    $\begingroup$ Presumably you wanted to do the following. Let $L/\Bbb{Q}$ be Galois with Galois group $A_4$. Then let $K\subset L$ be the fixed field of $\langle (123)\rangle$. Then $[K:\Bbb{Q}]=12/3=4$, and there are no intermediate fields, because $A_3$ is a maximal subgroup of $A_4$. $\endgroup$ – Jyrki Lahtonen Nov 12 '14 at 21:23

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