I'd appreciate a good clear reference for the proof of $1+2\mathbb Z_2\cong \mathbb Z/2\mathbb Z\times\mathbb Z_2$.
I've looked in "P-adic Numbers: An Introduction" by Gouvêa but couldn't find it.
Update:
Found the proof in Serres' "A Course in arithmetic" ch. 2 part 3 Prop. 8.
$$1+4\Bbb{Z}_2= (1+4)^{\Bbb{Z}_2}$$
Proof :
If $(a_n)\in \Bbb{Z}$ and $\lim_{n \to \infty} a_n$ converges in $\Bbb{Z}_2$ then $\lim_{n \to \infty} (1+4)^{a_n}$ converges in $1+4\Bbb{Z}_2$.
For all $n$ then $(1+4)^{2^n}=\sum_{k=0}^{2^n} {2^n\choose k} 4^k \equiv 1+ 2^{n+2} \bmod 2^{n+3}$
Given $b\in 1+4\Bbb{Z}_2$ by induction there exists $a_n\bmod 2^{n+1}$ such that $b \equiv (1+4)^{a_n} \bmod 2^{n+2}$ and hence $$b = \lim_{n \to \infty} (1+4)^{a_n} = (1+4)^{\lim_{n \to \infty} a_n}$$