What is a uniformizer?

It is a generator of the maximal ideal of a local field's valuation ring; for the p-adic numbers the prime p itself serves as a uniformizer, and every nonzero element is a unit times a power of it.

Why are the p-adic integers compact?

They are an inverse limit of the finite rings of integers modulo powers of p, which makes them a closed and bounded set in the p-adic metric and hence compact, unlike the ordinary integers.

p-adic Fields and Local Fields

The p-adic field is built by completing the rationals for the p-adic absolute value; its ring of p-adic integers, residue field, and uniformizer make it the model example of a local field, the natural home of arithmetic at a single prime.

Definition

The p-adic absolute value of a rational number is determined by the power of p dividing it. The field of p-adic numbers is the completion of the rationals under this absolute value; a local field is a field complete with respect to a discrete valuation and having a finite residue field.

Scope

This topic covers the p-adic valuation and absolute value, the ultrametric inequality, Ostrowski's classification of absolute values on the rationals, the construction of the p-adic numbers and the ring of p-adic integers, the maximal ideal, residue field, and uniformizer, the description of elements by p-adic digit expansions, Hensel's lemma for lifting roots, and the general notion of a local field as a complete discretely valued field with finite residue field.

Core questions

How is the p-adic absolute value defined, and why does it satisfy the strong ultrametric inequality?
Why does Ostrowski's theorem say these are essentially the only absolute values on the rationals besides the usual one?
What are the p-adic integers, and how do digit expansions and the residue field describe their structure?
How does Hensel's lemma lift solutions from the residue field to the full local field?

Key theories

Ostrowski's theorem and completions: Every nontrivial absolute value on the rationals is equivalent to the ordinary one or to a p-adic one; completing under each gives the reals or a p-adic field, exhibiting all places of the rationals.
Structure of p-adic integers: The p-adic integers form a compact local ring with maximal ideal generated by p and residue field the integers modulo p; every p-adic number has a unique base-p expansion possibly infinite to the right.
Hensel's lemma: A simple root of a polynomial modulo p lifts uniquely to a root in the p-adic integers; this makes the local field behave like an algebraically convenient enlargement of the residue field.

Clinical relevance

Local fields are the setting for local class field theory and for the local components of automorphic representations in the Langlands program; Hensel lifting is also an algorithmic tool in polynomial factorization and in fast computation modulo prime powers.

History

Hensel introduced p-adic numbers in 1897 to import power-series techniques into number theory, and proved the lifting lemma named for him. Ostrowski classified the absolute values on the rationals in 1916, clarifying that the real and p-adic completions exhaust the possibilities and grounding the local viewpoint.

Key figures

Kurt Hensel
Alexander Ostrowski
Helmut Hasse

Seminal works

serre1973
koblitz1984

Frequently asked questions

What is a uniformizer?: It is a generator of the maximal ideal of a local field's valuation ring; for the p-adic numbers the prime p itself serves as a uniformizer, and every nonzero element is a unit times a power of it.
Why are the p-adic integers compact?: They are an inverse limit of the finite rings of integers modulo powers of p, which makes them a closed and bounded set in the p-adic metric and hence compact, unlike the ordinary integers.