= Unit decomposition of the 2-adic Gaussian field
{title2=$\mathbb Z_2[i]^\times=\mu_4\times(1+(1-i)^3\mathbb Z_2[i])$}
The <Eisenstein polynomial> of $1-i$ is $T^2-2T+2$, so $\mathbb Q_2(i)$ has <residue field> $\mathbb F_2$ and <ramification index> two. The four roots $1,i,-1,-i$ have distinct cosets modulo $U_3$, which has index four by the <successive quotients of principal-unit groups>. The <logarithm isomorphism on deep principal units> gives $U_3\cong(\mathbb Z_2[i],+)$. Hence these four roots split the full unit group, and are all its torsion.
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