Let $p \neq 2$ a prime and consider the the field $\mathbb{Q}_p$ of $p$-adic numbers.
I have two questions:
How to derive following two isomorphisms (as multiplicative groups):
1) $\mathbb{Q}_p^* \cong \mathbb{Z} \times \mathbb{Z}^* _p$
2) $\mathbb{Q}_p^*/{\mathbb{Q}_p^*} ^2 \cong \mathbb{Z}/(2) \times \mathbb{Z}/(2)$
My ideas:
to 1):
Set theoretically we have $\mathbb{Q}_p = \bigcup _{n \in \mathbb{N}}1/p^n \mathbb{Z}_p$. Now what about $\mathbb{Q}_p^*$?
A concequence of Hensels lemma is that
$\mathbb{Z}_p ^* \cong \mathbb{Z}/(p-1) \times (1+p\mathbb{Z}_p) \cong \mathbb{Z}/(p-1) \times \mathbb{Z}_p = C_{p-1} \times \mathbb{Z}_p$
with $C_{p-1} := \mathbb{Z}/(p-1)$ as multiplicative group. Here $\mathbb{Z}/(p-1) $ are hothing but the lifted roots of $X^{p-1}-1$ in $\mathbb{F}_p$ (works by Hensel)
Then we observe that $\mathbb{Q}_p^*= \bigcup _{n \in \mathbb{Z}}p^n \mathbb{Z}_p^*$ since all $p^n \mathbb{Z}_p^*$ are pairwise disjoint for different $n \in \mathbb{Z}$.
Set theoretically this imply 1). But does this isomorphism also holds for these objects as groups?
to 2):
Using 1) provisionally we have $\mathbb{Q}_p^* \cong \mathbb{Z} \times \mathbb{Z}^* _p \cong \mathbb{Z} \times C_p \times \mathbb{Z} _p$.
Then ${\mathbb{Q}_p^*} ^2 = \cong 2\mathbb{Z} \times C_p^2 \times 2\mathbb{Z} _p$ (here I took into account if the group action is mutliplicative or additively written.
But I don't see why this implies $\mathbb{Q}_p^*/{\mathbb{Q}_p^*} ^2 \cong \mathbb{Z}/(2) \times \mathbb{Z}/(2)$.
Did I somewhere a mistake?
(1) Every element of $\mathbb{Q}_p^{\times}$ is of the form $p^nu$ where $n\in\mathbb{Z}$ and $u\in\mathbb{Z}_p^{\times}$, as you note. This means there is a bijection $\mathbb{Z}\times\mathbb{Z}_p^{\times}\to\mathbb{Q}_p^{\times}$. It's rather easy to check it's a group homomorphism; just check.
As for (2), you need to handle $C_{p-1}/C_{p-1}^2$ and $\mathbb{Z}_p/2\mathbb{Z}_p$ separately. Can you do that? It's also important to note that $(G_1\times G_2)/(H_1\times H_2)\cong (G_1/H_1)\times(G_2/H_2)$, but with three factor groups in your case.