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Let $K$ be a local field. Let $K^{nr}$ and $K^t$ be its maximal unramified and tamely ramified extensions, respectively. One can show that $\operatorname{Gal}(K^{nr}/K) \cong \widehat{\mathbb Z}$ and that $\operatorname{Gal}(K^{nr}/K)$ is topologically generated by the Frobenius element.

Theorem: $\operatorname{Gal}(K^t/K^{nr}) \cong \prod_{\ell \neq p} \mathbb Z_\ell$ and $\operatorname{Gal}(K^t/K) \cong \prod_{\ell \neq p} \mathbb Z_\ell \rtimes \widehat{\mathbb Z}$, where the action of $\widehat{\mathbb Z}$ is determined by the Frobenius acting by conjugation.

Kevin Buzzard cites this theorem in his lecture series on the Langlands programme (available on youtube). It should follow fairly easily from the Sylow theorems for profinite groups.

However I have not been able to find a reference for it in any textbooks! Can somebody please help me out?

Edit: The theorem is covered in Neukirch-Schmidt-Wingberg's "Cohomology of Number Fields" as Proposition 7.5.2. Credit goes to Tom Fisher.

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  • $\begingroup$ It may be worth noting that, as you progress with your mathematical career, you'll find more and more that the results you need aren't found in textbooks any more. To the best of my knowledge, there is no textbook introduction to Galois representations that would contain this fact. The best resource I'm aware of is Taylor's course in the 1990s (see p.20) and the various other courses based on it (including Buzzard's). You may find it in some other textbook that touches on Galois representations in some other way. $\endgroup$ – Mathmo123 May 2 '20 at 17:05
  • $\begingroup$ @Mathmo123 Thank you. As I noted in my edit, I ended up finding a reference. $\endgroup$ – Lukas Kofler May 2 '20 at 17:09
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As usual let $K$ be a finite extension of $\Bbb{Q}_p$. Then $$K^{nr} =\bigcup_{p\,\nmid \,n} K(\zeta_n), \qquad K^t =\bigcup_{p\,\nmid \,n} K(\pi_K^{1/n})=\bigcup_{p\,\nmid \,n} K(p^{1/n})$$

$$Gal(K^{t}/K^{nr}) = \{ (\pi_K^{1/n} \to \zeta_n^{a\bmod n} \pi_K^{1/n}), a\in \widehat{\Bbb{Z}}/\Bbb{Z}_p\}$$

$Gal(K^{nr}/K)$ is a finite index subgroup of $\widehat{\Bbb{Z}}^\times/\Bbb{Z}_p^\times$, choosing the natural Frobenius of $K^t$ gives an embedding $Gal(K^{nr}/K)\to Gal(K^{t}/K)$ from which you get your $\rtimes$.

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  • $\begingroup$ Thanks for answering my questions as always! I am looking for an actual reference in a textbook, though. $\endgroup$ – Lukas Kofler May 1 '20 at 21:56
  • $\begingroup$ Why, what is unclear to you ? By Hensel lemma all the $\bmod$-separable polynomials split in $K^{nr}$ and $K^t$ contain all its totally tamely ramified extensions from Hensel lemma again. Then the Galois group is supposedly obvious. $\endgroup$ – reuns May 1 '20 at 22:06
  • $\begingroup$ I would like to quote the theorem in a text I'm writing and I would like to put a reference for the proof. $\endgroup$ – Lukas Kofler May 1 '20 at 22:48
  • $\begingroup$ I have found an answer and accepted yours to take my questions off the "unanswered" list. $\endgroup$ – Lukas Kofler May 2 '20 at 11:29

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