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Is there a necessary and sufficient condition for when a cubic extension of $\mathbb{Q}$ is not a Galois extension?

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    It is sufficient that the extension is with a real root of $x^3 - m$, with $m \neq 0,1$ without cube factors. (Since the extension is not normal in this case). OK, I now see you wanted "iff" conditions, I didn't notice that until now. Sorry, I'll leave this comment anyway...2012-07-07

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