= Absolute Galois group of the complex Laurent series field
{title2=$\operatorname{Gal}(\overline{\mathbb C((t))}/\mathbb C((t)))\cong\widehat{\mathbb Z}$}
Inside an algebraic closure, choose compatible roots $t^{1/n}$. The fields $\mathbb C((t^{1/n}))$ exhaust all finite extensions by <finite extensions of the complex Laurent series field>. Restriction from a degree-$m$ field to a degree-$n$ field for $n\mid m$ sends an exponent modulo $m$ to its reduction modulo $n$. The <absolute Galois group> is their <inverse limit>, namely the <profinite completion> of $\mathbb Z$. This is a procyclic group, rather than the direct-limit torsion group of all complex roots of unity.
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