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## Homework Statement

If p does not divide a, show that [tex]a_n=a^{p^{n}}[/tex] is Cauchy in [tex]\mathbb{Q}_p[/tex].

## The Attempt at a Solution

We can factor [tex]a^{p^{n+k}}-a^{p^n}=a^{p^n}(a^{p^{n+k}-1}-1)[/tex]. p doesn't divide [tex]a^{p^n}[/tex] so somehow I must show that [tex]a^{p^{n+k}-1}-1[/tex] is divisible by larger and larger powers of p. I feel it has something to do with the totient theorem, but I can't get it to work.

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