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What is the Galois Group of $\mathbb{Q}_p(\zeta_p,\sqrt[p]{p})$ where $\zeta_p$ is the primitive p-th root of unity. I know it is a totally ramified extension of degree $p(p-1)$. However, how can I find the Galois group of this extension?

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  • $\begingroup$ You may want to consider Hensel's lemma, www1.spms.ntu.edu.sg/~frederique/antchap7.pdf $\endgroup$ – Chickenmancer Apr 21 '17 at 2:14
  • $\begingroup$ Could you please give a little bit more detail? I am not sure how would Hensel's lemma help since this is totally ramified extension, hence the residue field extension is trivial. $\endgroup$ – Jiali Yan Apr 21 '17 at 2:50
  • $\begingroup$ Page 12 of the PDF (not the book the PDF is of) provides a treatment of totally ramified extensions. $\endgroup$ – Chickenmancer Apr 21 '17 at 2:56
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    $\begingroup$ This has nothing to do with Hensel's Lemma, since the Galois group is the same if you replace the p-adic field by the rationals. The group is a Frobenius group, and you can determine its structure by writing down two automorphisms that generate the whole group. One of them is the trivial lift of the automorphism of the cyclotomic extension, the other maps a p-th root to its conjugate. $\endgroup$ – franz lemmermeyer Apr 21 '17 at 6:34

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