Why this sequence $1 \rightarrow \mathbb{U}_1/\mathbb{U}_n \rightarrow \mathbb{U}/\mathbb{U}_n \rightarrow \mathbb{F}_p^* \rightarrow 1$ is exact?

Question

I'm trying to understand the algebraic properties of p-adic numbers in the "algebraic" part of Serre' book "A Course in arithmetic". In the chapter 2 section 3.

$\mathbb{U} := \mathbb{Z}_p^*$ the units of the p-adic integers, $\mathbb{U}_n := 1 +p^n\mathbb{Z}_p$ and $\mathbb{F}_p^* = \mathbb{Z} / p\mathbb{Z}^*$.

He says that this sequence $1 \rightarrow \mathbb{U}_1/\mathbb{U}_n \rightarrow \mathbb{U}/\mathbb{U}_n \rightarrow \mathbb{F}_p^* \rightarrow 1$ is exact, but

he doesn't say what are the maps. What are the maps involved?

And, why these maps implies the exactness of the sequence?

Answer 1

The first map is the inclusion $i: \mathbb{U}_1 \rightarrow \mathbb{U}$ and the second map is the reduction mod $p$ $\pi: \mathbb{U} \rightarrow \mathbb{F}_p^{\times}$.

The exactness follows because you’re quotienting the exact sequence $1 \rightarrow \mathbb{U}_1 \rightarrow \mathbb{U} \rightarrow \mathbb{F}_p^{\times}$ by a subgroup of the kernel of the surjection.