🔢Algebraic Number Theory Unit 11 – Local Fields and Completions

Key Concepts and Definitions

• Local fields finite field extensions of $\mathbb{Q}_p$ (p-adic numbers) or $\mathbb{F}_p((t))$ (formal power series over a finite field)
• Completions process of extending a field with respect to a given absolute value or norm, resulting in a complete metric space
• Absolute values functions that assign a non-negative real number to each element of a field, satisfying certain properties (multiplicative, non-Archimedean)
• Norms generalizations of absolute values, assigning a non-negative real number to each element of a vector space (satisfy triangle inequality, positive definiteness)
• Hensel's Lemma powerful tool for lifting solutions of polynomial equations from a residue field to a complete local field
  • Analogous to Newton's method in calculus
• Unramified extensions local field extensions with the same residue field as the base field
• Totally ramified extensions local field extensions where the residue field extension is trivial

Absolute Values and Norms

• Absolute values functions $|\cdot|: K \to \mathbb{R}_{\geq 0}$ satisfying:
  1. $|x| = 0$ if and only if $x = 0$
  2. $|xy| = |x||y|$ for all $x, y \in K$
  3. $|x + y| \leq |x| + |y|$ for all $x, y \in K$ (triangle inequality)
• Non-Archimedean absolute values satisfy a stronger triangle inequality: $|x + y| \leq \max(|x|, |y|)$
• Trivial absolute value $|x| = 1$ for all $x \neq 0$ and $|0| = 0$
• p-adic absolute value for a prime $p$, $|x|_p = p^{-\text{ord}_p(x)}$, where $\text{ord}_p(x)$ is the highest power of $p$ dividing $x$
• Norms generalizations of absolute values to vector spaces over a valued field
  • Satisfy positive definiteness, homogeneity, and triangle inequality
• Equivalence of norms two norms are equivalent if they induce the same topology on the vector space

p-adic Numbers and Fields

• p-adic numbers $\mathbb{Q}_p$ completion of $\mathbb{Q}$ with respect to the p-adic absolute value $|\cdot|_p$
  • Elements of $\mathbb{Q}_p$ can be represented as power series in $p$ with coefficients in $\{0, 1, \ldots, p-1\}$
• p-adic integers $\mathbb{Z}_p$ subring of $\mathbb{Q}_p$ consisting of elements with non-negative p-adic valuation
• Unit ball $\{x \in \mathbb{Q}_p : |x|_p \leq 1\}$ is equal to $\mathbb{Z}_p$
• Residue field of $\mathbb{Q}_p$ is the finite field $\mathbb{F}_p$
• Hensel's Lemma applies to lifting solutions of polynomial equations from $\mathbb{F}_p$ to $\mathbb{Z}_p$
• Extensions of $\mathbb{Q}_p$ can be classified as unramified or totally ramified

Completion of Fields

• Completion process of extending a field $K$ with respect to an absolute value $|\cdot|$ to obtain a complete metric space $\hat{K}$
  • Elements of $\hat{K}$ are equivalence classes of Cauchy sequences in $K$
• Completion is unique up to isometric isomorphism
• Completions with respect to equivalent absolute values are isomorphic
• Completion of $\mathbb{Q}$ with respect to the p-adic absolute value is $\mathbb{Q}_p$
• Completion of $\mathbb{F}_p(t)$ with respect to the t-adic absolute value is $\mathbb{F}_p((t))$
  • Elements of $\mathbb{F}_p((t))$ are formal power series in $t$ with coefficients in $\mathbb{F}_p$
• Completions are important in the study of local fields and their extensions

Local Fields: Properties and Examples

• Local fields complete valued fields with a discrete valuation and a finite residue field
  • Characteristic 0 local fields finite extensions of $\mathbb{Q}_p$
  • Characteristic $p$ local fields finite extensions of $\mathbb{F}_p((t))$
• Discrete valuation a surjective homomorphism $v: K^\times \to \mathbb{Z}$ satisfying $v(x + y) \geq \min(v(x), v(y))$
• Valuation ring $\mathcal{O}_K = \{x \in K : v(x) \geq 0\}$, a local ring with maximal ideal $\mathfrak{m}_K = \{x \in K : v(x) > 0\}$
• Residue field $k_K = \mathcal{O}_K / \mathfrak{m}_K$, a finite field
• Examples of local fields:
  • $\mathbb{Q}_p$, $\mathbb{F}_p((t))$
  • Finite extensions of $\mathbb{Q}_p$ (e.g., $\mathbb{Q}_p(\sqrt{d})$ for $d \in \mathbb{Z}$)
  • Finite extensions of $\mathbb{F}_p((t))$ (e.g., $\mathbb{F}_p((t^{1/n}))$ for $n \in \mathbb{N}$)
• Local fields have a compact topology induced by their absolute value

Hensel's Lemma and Applications

• Hensel's Lemma a powerful tool for lifting solutions of polynomial equations from the residue field to the valuation ring of a local field
  • Analogous to Newton's method in calculus
• Statement: Let $(K, v)$ be a complete valued field with valuation ring $\mathcal{O}_K$ and residue field $k_K$. Let $f(x) \in \mathcal{O}_K[x]$ be a polynomial and $\bar{a} \in k_K$ be a simple root of $\bar{f}(x)$ (the reduction of $f$ modulo the maximal ideal). Then there exists a unique $a \in \mathcal{O}_K$ such that $f(a) = 0$ and $a \equiv \bar{a} \pmod{\mathfrak{m}_K}$.
• Applications:
  • Solving polynomial equations over local fields
  • Constructing extensions of local fields
  • Proving the Local-Global Principle (Hasse Principle) for certain Diophantine equations
• Hensel's Lemma can be generalized to systems of polynomial equations (multivariate Hensel's Lemma)

Extensions of Local Fields

• Extension theory of local fields analogous to the extension theory of global fields (number fields, function fields)
• Unramified extensions local field extensions with the same residue field as the base field
  • Degree of an unramified extension equals the degree of the residue field extension
• Totally ramified extensions local field extensions where the residue field extension is trivial
  • Degree of a totally ramified extension equals the ramification index (the index of value groups)
• Fundamental equality for a finite extension $L/K$ of local fields: $[L:K] = e(L/K)f(L/K)$, where $e(L/K)$ is the ramification index and $f(L/K)$ is the residue field extension degree
• Local Kronecker-Weber Theorem every finite abelian extension of $\mathbb{Q}_p$ is contained in a cyclotomic extension $\mathbb{Q}_p(\zeta_{p^n})$ for some $n$
• Structure of the absolute Galois group of a local field (e.g., $\text{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$) is known and simpler than the absolute Galois group of a global field

Connections to Global Fields

• Local-Global Principle (Hasse Principle) a method of solving Diophantine equations by reducing the problem to solving equations over local fields
  • A Diophantine equation has a solution over a global field if and only if it has a solution over every completion of the global field
• Adeles and Ideles constructions that allow the study of global fields using the properties of their completions
  • Adeles the restricted direct product of the completions of a global field with respect to all absolute values (or valuations)
  • Ideles the multiplicative group of the adeles
• Hasse-Minkowski Theorem a quadratic form over a global field has a non-trivial zero if and only if it has a non-trivial zero over every completion of the global field
• Hasse Norm Theorem an element of a global field is a norm from a cyclic extension if and only if it is a norm locally everywhere
• Connections between local and global class field theory, which describes abelian extensions of global fields in terms of the arithmetic of the base field