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February 1st, 2018, 01:46 AM   #1
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Cyclic normal group

Please help me to prove the following problem:

Let $f\subset E$ be a Galois extension and assume that $E=F(\alpha)$ with $\alpha^n\in F$. Let $G=Gal(E/F)$. Show that $G$ has a cyclic normal subgroup $N$ of order dividing $n$ such that $G/N$ is abelien and has order dividing $\phi(n)$.

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