1. Feb 24, 2007

### Dragonfall

1. The problem statement, all variables and given/known data

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

3. The attempt at a solution

We can factor $$a^{p^{n+k}}-a^{p^n}=a^{p^n}(a^{p^{n+k}-1}-1)$$. p doesn't divide $$a^{p^n}$$ so somehow I must show that $$a^{p^{n+k}-1}-1$$ 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.

Last edited: Feb 24, 2007
2. Feb 25, 2007

Anyone help?