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So assume we are given some $a\in\mathbb{Z}_p^\times$ and we want to figure out if $X^p-a$ has a root in $\mathbb{Q}_p$. We know that such a root must be unique, because given two such roots $\alpha,\beta$, the quotient $\alpha/\beta$ would need to be a non-trivial $p^\textrm{th}$ root of unity and $\mathbb{Q}_p$ does not contain any.

Now we can't apply Hensel, which is the canonical thing to do when looking for roots in $\mathbb{Q}_p$. What other approaches are available?

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    Start with the direct sum decomposition: $$\mathbf{Q}_p^*\simeq \langle p\rangle\times\mu_{p-1}\times U_1,$$ where $U_1$ consist of $p$-adic integers congruent to 1 modulo $p$. The first two factors are easy. The last one you need to work on.2011-09-21

3 Answers 3