Monday, September 06

$k^{\mathrm{perf}}\coloneqq\displaystyle\bigcup_{n\geq 1} k^{1\over p^n} \subseteq \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu$.

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!

Archimedean fields: ${\mathbb{R}}, {\mathbb{C}}$. Everything else is nonarchimedean.

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.

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 $\dim_{\krull}(A) = 1$.

Function field: an extension $F_{/ {k}}$ where $[F: k(x)] < \infty$ for some $x$ transcendental over $k$.

DVR: a local PID that is not a field.

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.