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I found a short problem while reviewing some material.

True/False: The splitting field over $\mathbb Q$ of the polynomial $x^{17} − 5 \in \mathbb Q[x]$ has a cyclic Galois group.

It's clear that this splitting field is a Galois extension, but I'm not sure how to solve the problem. Any help would be great.

CuriousKid7
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  • The extension $\mathbb{Q}(\sqrt[17]{5})/\mathbb{Q}$ is not Galois, so the Galois group of the splitting field of the polynomial cannot be abelian, thus cannot be cyclic. – Lukas May 18 '17 at 09:07
  • @Crostul The product is semidirect. – Lukas May 18 '17 at 09:09
  • @Lukas Why is that not Galois? – CuriousKid7 May 18 '17 at 09:37
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    @CuriousKid7 the field $\mathbb{Q}(\sqrt[17]{5})$ can be embedded into $\mathbb{R}$, but $x^{17}-5$ clearly has complex roots, hence it cannot contain all of them. – Matt B May 18 '17 at 10:15

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