absolute Galois group

Definition: For \(k\in \mathsf{Field}\), the absolute Galois group is
\begin{align*} G_k \coloneqq{ \mathsf{Gal}} (k_s/k) \cong \mathop{\mathrm{Aut}}(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu/k) .\end{align*}
- Warning: \(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu/k\) may not be Galois! Hence the need for a separable closure.
#why-care: \({ \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbf{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbf{Q}})\) generalizes class field theory and packages together all finite extensions of \({\mathbf{Q}}\).
For \(K^{\operatorname{ab}}\) the maximal abelian extension of \(K\), finite abelian extensions of \(K\) correspond to open subgroups of \({ \mathsf{Gal}} (K^{\operatorname{ab}}/K)\), which are finite index since this group is compact.

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