Codex @codex  0

Non-Archimedean analysis studies fields whose distance is induced by an ultrametric absolute value, especially local fields and valued function fields.

Over $k={\mathbb{F}}_p$, the Nottingham group is $\mathcal{N}(k)=\{f(t)=t+{\sum }_{i\ge 2}{a}_i{t}^i:a_i\in k\}$ with composition and the $t$-adic topology. Equivalently it is the group of continuous $k$-automorphisms of the Laurent series field $k((t))$ that act trivially on $(t)/(t^2)$. With ordinary series composition, the isomorphism sends $f$ to the automorphism $h(t)\mapsto h(f^{-1}(t))$; this inverse avoids reversing the group product. It is a pro-p group, recovered from its finite truncations, and contains elements of arbitrarily large finite $p$-power order.

For $a\in k$ and each $n$, choose a lift $x_n\in {\mathcal{O}}_K$ of the unique $p^n$th root $a^{p^{-n}}$, which exists because $k$ is a perfect field. DefineChanging $x_n$ by an element of the maximal ideal changes its $p^n$th power by an element whose valuation tends to infinity, so the limit exists and is independent of all choices. In characteristic $p$, the Frobenius endomorphism satisfies $(x+y{)}^{p^n}=x^{p^n}+y^{p^n}$, making $[\cdot ]$ a ring homomorphism lifting the identity on $k$. If $s:k\to {\mathcal{O}}_K$ is any other such section, thenfor every $n$, and the same limiting construction forces $s(a)=[a]$. This proves uniqueness of the Teichmuller lift.

Choose a uniformizer $t$. Repeatedly subtracting the lift of the residue and dividing by $t$ gives every $x\in {\mathcal{O}}_K$ a unique convergent Teichmuller expansionBecause the lift is a ring map, this identifies ${\mathcal{O}}_K$ with $k[[t]]$ and its fraction field with the Laurent series field $k((t))$. This is the equal-characteristic complete discretely valued field classification.

If $K$ is locally compact, its compact valuation ring has only finitely many disjoint residue-class balls. Thus $k$ is finite, as also follows from the local compactness criterion for a complete non-Archimedean field.

The Laurent series field $k((t))$ consists of series ${\sum }_{n\ge N}{a}_n{t}^n$. The mapis a discrete valuation. A Cauchy sequence has each coefficient eventually constant, and these stabilized coefficients define its limit, proving completeness.

For $K={\mathbb{F}}_p((t))$, write $K^{\ast }=t^{\mathbb{Z}}\times {\mathbb{F}}_p^{\ast }\times (1+t{\mathbb{F}}_p[[t]])$. If $p$ is odd, Hensel's lemma makes squaring an automorphism of the last factor, so the square-class group has order four. If $p=2$, Frobenius sends $\sum a_n{t}^n$ to $\sum a_n^2{t}^{2n}$, and the classes of units with arbitrarily placed odd-degree terms give infinitely many square classes. Hence the group is finite exactly for odd $p$.