Exercise 24, § 6

a) Show that every subgroup of the multiplicative group $\mathbf{Q}_p^*$ of the $p$-adic field $\mathbf{Q}_p$ is isomorphic to the product of a subgroup of the multiplicative group $U$ of units of $\mathbf{Z}_p$ and a discrete additive subgroup isomorphic to $\mathbf{Z}$ or $\{0\}$.

b) Show that the compact subgroups of the subgroup $V = 1 + \mathfrak{p}$ of $U$ are the groups $1 + \mathfrak{p}^n$ [same reasoning as in Exercise 23 c)].

c) Show that, for each $a \in U$, the sequence $(a^{p^n})_{n \in \mathbf{N}}$ tends to a limit $\alpha \equiv a \pmod{\mathfrak{p}}$ and that $\alpha^p = \alpha$ [show that $a^{p^n} \equiv a^{p^{n-1}} \pmod{\mathfrak{p}^n}$ by induction on $n$]. Show that all the roots of the polynomial $X^{p-1} - 1$ (in an algebraically closed extension of $\mathbf{Q}_p$) belong to $\mathbf{Q}_p$ and are pairwise incongruent mod $\mathfrak{p}$ [apply what precedes to the roots of the congruence $x^{p-1} - 1 \equiv 0 \pmod{p}$]. If $p > 2$ and if $d$ is the highest common factor of $n$ and $p - 1$, the polynomial $X^n - 1$ has exactly $d$ roots in $\mathbf{Q}_p$, which are roots of $X^d - 1$ (same method : take $a$ to be a root of $X^n - 1$).

Hence show that, if $p > 2$, every compact subgroup of the multiplicative group $\mathbf{Q}_p^*$ is the direct product of a finite subgroup of $U$ [consisting of the $(p-1)$th roots of unity] and a subgroup of the form $1 + \mathfrak{p}^n$ [use b)]. How are these results to be modified for $p = 2$? (Make the group $1 + \mathfrak{p}^2$ play the part previously played by $1 + \mathfrak{p}$).