Local Fields and Valuations - Key Properties

Local field extensions have remarkably regular structure, classified by ramification data and described explicitly via Eisenstein polynomials.

TheoremHensel's Lemma

Let $K$ be a complete discrete valuation field with residue field $\kappa$ . If $f(x) \in \mathcal{O}_K[x]$ and $\bar{a} \in \kappa$ satisfies $\bar{f}(\bar{a}) = 0$ and $\bar{f}'(\bar{a}) \neq 0$ in $\kappa[x]$ , then there exists unique $a \in \mathcal{O}_K$ with $f(a) = 0$ and $\bar{a}$ the reduction of $a$ .

Consequence: Simple roots modulo $\mathfrak{m}$ lift uniquely to actual roots. This enables solving polynomial equations $p$ -adically via approximation.

ExampleSquare Roots in $\mathbb{Q}_p$

For which $a \in \mathbb{Z}_p$ does $x^2 = a$ have solutions in $\mathbb{Q}_p$ ?

By Hensel: if $a \equiv b^2 \pmod{p}$ for some $b$ coprime to $p$ , then $\sqrt{a}$ exists in $\mathbb{Z}_p$ .

For $p$ odd: squares in $\mathbb{Q}_p^*$ are characterized by $v_p(a)$ even and $a/p^{v_p(a)} \in (\mathbb{Z}_p^*)^2$ .

For $p = 2$ : more subtle, requiring $a \equiv 1 \pmod{8}$ when $v_2(a) = 0$ .

DefinitionUnramified and Totally Ramified Extensions

An extension $L/K$ of local fields is:

Unramified if $e = 1$ and the residue extension $\ell/\kappa$ is separable
Totally ramified if $f = 1$ (no change in residue field)

Every finite extension $L/K$ has a unique decomposition: $L/L_0/K$ where $L_0/K$ is maximal unramified and $L/L_0$ is totally ramified.

TheoremStructure of Extensions

Let $K$ be a local field with residue field $\mathbb{F}_q$ .

Unramified extensions: Unique extension of degree $n$ for each $n \geq 1$ , obtained by adjoining roots of $x^{q^n} - x$
Totally ramified extensions: Generated by roots of Eisenstein polynomials $f(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_0$ with $v(a_i) > 0$ and $v(a_0) = 1$
Maximal unramified extension: $K^{nr}$ has residue field $\bar{\mathbb{F}}_q$ (algebraic closure of $\mathbb{F}_q$ )

The absolute Galois group satisfies: $\text{Gal}(\bar{K}/K) \twoheadrightarrow \text{Gal}(K^{nr}/K) \cong \hat{\mathbb{Z}}$ (profinite integers).

ExampleExtensions of $\mathbb{Q}_p$

Unramified degree 2: $\mathbb{Q}_p(\sqrt{a})$ where $a \in \mathbb{Z}_p^*$ is not a square mod $p$
Totally ramified degree 2: $\mathbb{Q}_p(\sqrt{p})$
Mixed ramification: $\mathbb{Q}_p(\zeta_p)$ when $p \neq 2$ is totally ramified of degree $p-1$

For $\mathbb{Q}_2$ : the extension $\mathbb{Q}_2(i)$ has $e = 2, f = 1$ , while $\mathbb{Q}_2(\sqrt{3})$ has $e = 1, f = 2$ .

Remark

The regularity of local extensions contrasts with global fields. Every local extension is solvable (Lubin-Tate theory constructs explicit totally ramified abelian extensions). This simplicity makes local class field theory explicit and computable.