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Definition : Let $K$ be a field. If $\alpha$ is an algebraic element over K, such that $\alpha^n \in K $ and such that $x^n-\alpha^n \in K[x] $ is irreducible over K. Then we call $ K(\alpha)$ a simple radical and irreducible extension. A field extension $ L / K$ is said to be solvable by simple and irreducible radicals, if there exist a simple radical and irreducible extension $K(\alpha)$ such that $ K(\alpha) / L $.

Problem:

Let $\zeta_n$ be a primitive n-rooth of unity. Prove that the field extension $ \Bbb Q(\zeta_n) $ is solvable by simple and irreducible radicals.

And then give a field tower of simple and irreducible radicals extensions $\Bbb Q = E_0 \subset E_1 \subset ... \subset E_n $ such that $ \zeta_{47}\in E_n $

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    I think nobody will ever be able to prove the first assertion unless you edit it appropriately.2012-11-25

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