I am trying to understand the structure of the $p$-adic units. I know that we can write $$\mathbb{Z}_p^\times \cong \mu_{p-1} \times 1 + p\mathbb{Z}_p,$$ where $\mu_n$ are the $n$th roots of unity in $\mathbb{Z}_p$. My question is: what more can we say about the structure of $1 + p\mathbb{Z}_p$? It seems to be topologically generated by $1 + p$, right?
The reason I am interested is in order to understand $\mathbb{Z}_p^\times/\mathbb{Z}_p^{\times 2}$.
For $p$ odd, the $p$-adic exponential in the form $x\rightarrow \exp(px)$ provably gives a topological group isomorphism from $\mathbb Z_p$ with addition to $1+p\mathbb Z_p$ (the logarithm gives an inverse).
For $p=2$, tweaking is required.