I tend to ask questions on mathstackexchange, because I don't feel adequate yet for mathoverflow. I had previously asked this question there (I have now deleted it), where it was quite popular, but it didn't seem like anybody knew the answer. I thought that perhaps this question should graduate to mathoverflow. Here it is, as it had originally appeared:


I've been playing around with the $p$-adics, and I wondered about the structure of their Galois group.


We have the short exact sequence:


$$1\rightarrow Gal(\bar{\mathbb{Q}}\_p/\mathbb{Q}\_p^{un})\rightarrow Gal(\bar{\mathbb{Q}}\_p/\mathbb{Q}\_p)\rightarrow Gal(\mathbb{Q}\_p^{un}/\mathbb{Q}\_p)\rightarrow 1$$


I as wondering if this exact sequence is split. I.e., is it true that you can lift the Frobenius automorphism of $Gal(\mathbb{Q}\_p^{un}/\mathbb{Q}\_p)$ to $Gal(\bar{\mathbb{Q}}\_p/\mathbb{Q}\_p)$? How would the lifting of this automorphism act on individual elements of $\bar{\mathbb{Q}}\_p$?



Lifting the Frobenius to the absolute Galois group of the $p$-adics
As others have mentioned, the exact sequence is continuously split. Furthermore, any lift of Frobenius will act as you expect on the subfield $\mathbb{Q}\_p^{nr}$, i.e., the extension you get by adjoining all prime-to-$p$ roots of unity. Indeed, you can form a $\mathbb{Q}\_p$-basis of $\mathbb{Q}\_p^{nr}$ using roots of unity, and then the action is determined by $p$-th powers on the basis elements.


As far as other elements are concerned, the set of lifts is a torsor under $\operatorname{Gal}(\overline{\mathbb{Q}\_p}/\mathbb{Q}\_p^{nr})$, so you have a large amount of freedom. For example, your typical $S\_3$-extension whose intersection with $\mathbb{Q}\_p^{nr}$ is quadratic will have three Frobenius lifts.


