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I have a question concerning definition in terms of minimal polynomial i.e. if we let $E = F(\alpha)$ be a field extension of $F$ of degree two then how do I describe, in terms of the minimal polynomial for $\alpha$ over $F$ when this field extension is Galois?

Also does there exist a field extension of $\mathbb{Q}$ of degree $3$ that is Galois?

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    For your last question, have a look at this: http://sbseminar.wordpress.com/2008/04/21/classifying-z3-extensions-of-the-rationals/2012-05-21
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    Are you still here, Karl?2012-05-27

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