Deduce that $\mathbb{Z}_p^\times$ is isomorphic to $(\mathbb{Z}_p, +) \times \mathbb{F}_p^\times$. I have already deduced that $(1+p\mathbb{Z}_p, \cdot)$ and $(\mathbb{Z}_p,+)$ are isomorph. But I can't find the wanted isomorphism. Can someone help me?
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1What are $(Z)^\times _p$ and $(F)_p^\times $ ? – Surb Apr 15 '20 at 15:05
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2Teichmuller lift? – Angina Seng Apr 15 '20 at 15:29
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By the way, "deduce" from what? This words relates back to a previous result. – Torsten Schoeneberg Apr 15 '20 at 15:50
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Hint: You should know a map $\mathbb Z_p \twoheadrightarrow \mathbb F_p$. Does it induce a map on the respective units? Then, what is the kernel of that induced map? Then, does it split, i.e. is there a subgroup in the $p$-adic units which is cyclic of order $p-1$ (think about what that would mean for a $p$-adic number: It is a root of ...)?
As an aside, the isomorphism that you claim to already have is not true for $p=2$ (and is a more intricate result than all my hints above: It is a logarithm map).
As another aside, this question has been asked here before (check Units of p-adic integers and further duplicates linked there), and the isomorphism(s) are treated in each and every source on $p$-adics.
Torsten Schoeneberg
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