p- adic isomorphims

Question

1. Let $m_p=\{ x \in \mathbb{Q} : v_p(x)>0\}$, $v_p$ $p$-adic valuation. Let $k >0 $ a integer and the ideal $I=m_p^k$. Then, $$O_p/ I \cong \mathbb{Z}/p^k \mathbb{Z} $$ with, $O_p=\{ x \in \mathbb{Q} : v_p(x)\geq0\}$.

I have shown with $k=1$, but it does not work the induction...

1. Let $I_p =\{ x \in \mathbb{Z}_p : |x|_p <1\}\subset \mathbb{Z}_p$, a maximal ideal. Show that $$\mathbb{Z}_p/ I_p \cong \mathbb{Z}/p \mathbb{Z}$$ with, $|x|_p=(\frac{1}{p})^{v_p(x)}$

Thanks!

Answer 1

Hints:

1. You should note that $m_p=p\mathbb{Z}_p$, and thus $I=p^k\mathbb{Z}_p$. So, you're looking at $\mathbb{Z}_p/p^k\mathbb{Z}_p$. Try to find an isomorphism similar to the one from your previous question!
2. $|x|_p<1\Leftrightarrow v_p(x)>0$, and thus $I_p$ secretly just (BLANK) from 1., in disguise!