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For $p\neq 2$ it's easy to prove through the log/exp-correspondence that

$$(1+p\mathbb{Z}_p)^{p^k}=1+p^{k+1}\mathbb{Z}_p.$$

This gives an easy way to compute the groups $\mathbb{Q}_p^*/\mathbb{Q}_p^{*n}$ which are useful in a lot of computations involving local class field theory. I was wondering if there's some nice way to express the above equation in the case $p=2$?

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    The log/exp correspondence for $p=2$ involves $4{\mathbf Z}_2$ and $1 + 4{\mathbf Z}_2$. Start with $(1+2{\mathbf Z}_2)^2 = ({\mathbf Z}_2^\times)^2 = 1 + 8{\mathbf Z}_2 = 1 + 2\cdot 4{\mathbf Z}_2$.2011-11-11

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