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\(k^{\mathrm{perf}}\coloneqq\displaystyle\bigcup_{n\geq 1} k^{1\over p^n} \subseteq \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\).
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Absolute Galois group: \(G_k \coloneqq{ \mathsf{Gal}} (k_s/k) \cong \mathop{\mathrm{Aut}}(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu/k)\).
- It’s possible for \(\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu/k\) to not be Galois!
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Archimedean fields: \({\mathbb{R}}, {\mathbb{C}}\). Everything else is nonarchimedean.
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Local field:
- Finite extension of \({\mathbb{R}}, {\mathbb{Q}}, { {\mathbb{Q}}_p }, {\mathbb{F}}_p((t))\).
- \({\mathbb{R}}, {\mathbb{C}}, {\mathbb{F}}_{p^k}((t))\), or a finite extension of \({ {\mathbb{Q}}_p }\).
- \({\mathbb{R}}, {\mathbb{C}}, \operatorname{ff}(R)\) for \(R\) a complete DVR with finite residue field
- \(k\) a nondiscrete locally compact Hausdorff topological ring.
- \(k\) the completion of a global field with respect to a nontrivial absolute value.
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Global field: a number field (finite extension of \({\mathbb{Q}}\)) or a global function field.
- Global function field: a finite extension of \({\mathbb{F}}_p(t)\), or the function field of a geometrically integral curve over \({\mathbb{F}}_{p^k}\)
- Equivalently: \(\operatorname{ff}(A)\) for \(A\in {\mathsf{Alg}}_{/ {{\mathbb{Z}}}} ^{{\mathrm{fg}}}\) with \(A\) an integral domain and \(\dim_{\krull}(A) = 1\).
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Function field: an extension $F_{/ {k}} $ where \([F: k(x)] < \infty\) for some \(x\) transcendental over \(k\).
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DVR: a local PID that is not a field.
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Place:
- For function fields, maximal ideals \(p\) of some valuations rings \({\mathcal{O}}\). If \(p = \left\langle{t}\right\rangle = t{\mathcal{O}}\), then \(t\) is a uniformizer.
- Equivalence classes of valuations.