### Context

#### Algebra

higher algebra

universal algebra

# $p$-adic numbers

## Idea

For $p$ any prime number, the $p$-adic numbers $\mathbb{Q}_p$ are a field that completes the field of rational numbers. As such they are analogous to real numbers. A crucial difference is that the reals form an archimedean field, while the $p$-adic numbers form a non-archimedean field.

$p$-Adic numbers play a role in non-archimedean analytic geometry that is analogous to the role of the real numbers/Cartesian spaces in ordinary differential geometry.

There have long been speculations (see the references below) that this must mean that $p$-adic numbers also play a central role in the description of physics, see p-adic physics.

## Definition

Let $\mathbf{Z}$ be the ring of integers and for every $q\neq 0$, $q\mathbf{Z}$ its ideal consisting of all integer multiples of $q$, and $\mathbf{Z}/q\mathbf{Z}$ the corresponding quotient, the ring of residues mod $q$.

Let now $p\in \mathbf{Z}_+$ be a prime number. Then for any two positive integers $n\geq m$ there is an inclusion $p^m \mathbf{Z}\subset p^n\mathbf{Z}$ which induces the canonical homomorphism of quotients $\phi_{n,m}:\mathbf{Z}/p^n\mathbf{Z}\to \mathbf{Z}/p^m\mathbf{Z}$. These homomorphism for all pairs $n\geq m$ form a family closed under composition, and in fact a category, which is in fact a poset, and moreover a directed system of (commutative unital) rings. The ring of p-adic integers $\mathbf{Z}_p$ is the (inverse) limit of this directed system (inside the category of rings).

Regarding that the rings in the system are finite, it is clear that the underlying set of $\mathbf{Z}_p$ has a natural topology as a profinite (Stone) space and it is in particular a compact Hausdorff topological ring. More concretely, $\mathbf{Z}_p$ is the closed (hence compact) subspace of the cartesian product $\prod_{n} \mathbf{Z}/p^n\mathbf{Z}$ of discrete topological spaces $\mathbf{Z}/p^n\mathbf{Z}$ (which is by the Tihonov theorem compact Hausdorff) consisting of threads, i.e. sequences of the form $x = (...,x_n,...,x_2,x_1)$ with $x_n\in p^n\mathbf{Z}$ and satisfying $\phi_{n,m}(x_n) = x_m$.

The kernel of the projection $pr_n: \mathbf{Z}_p\to\mathbf{Z}/p^n\mathbf{Z}$, $x\mapsto x_n$ to the $n$-th component (which is the corresponding projection of the limiting cone) is $p^n\mathbf{Z}_p\subset\mathbf{Z}_p$, i.e. the sequence

$0 \longrightarrow \mathbf{Z}_p\stackrel{p^n}\longrightarrow \mathbf{Z}_p\longrightarrow \mathbf{Z}/p^n\mathbf{Z} \longrightarrow 0$

is an exact sequence of abelian groups, hence also $\mathbf{Z}_p/p^n\mathbf{Z}_p\cong \mathbf{Z}/p^n\mathbf{Z}$.

An element $u$ in $\mathbf{Z}_p$ is invertible (and called a $p$-adic unit) iff $u$ is not divisible by $p$.

Let $U\subset\mathbf{Z}_p$ be the group of all invertible elements in $\mathbf{Z}_p$. Then every element $x\in \mathbf{Z}_p$ can be uniquely written as $s= up^n$ with $n\geq 0$ and $u\in U$. The correspondence $x\mapsto n$ defines a discrete valuation $v_p:\mathbf{Z}_p\to \mathbf{Z}\cup\{\infty\}$ called the $p$-adic valuation and $n$ is said to be the $p$-adic valuation of $x$. Of course, $v_p(0)=\infty$ as required by the axioms of valuation. The metric induced by the valuation is (up to equivalence) given by

$d(x,y) = e^{-v_p(x-y)},$

ring $\mathbf{Z}_p$ is a complete metric space in that $d$ is a metric, and $\mathbf{Z}$ is dense in it.

The field of $p$-adic numbers $\mathbf{Q}_p$ is the field of fractions of $\mathbf{Z}_p$. The $p$-adic valuation $v_p$ extends to a discrete valuation, also denoted $v_p$ on $\mathbf{Q}_p$. Indeed, it is still true for all $x\in \mathbf{Q}_p$ that they can be uniquely written in the form $p^n u$ where $u\in U$ (the same group $U$ as before), but now one needs to allow $n\in \mathbf{Z}$. One defines the metric on $\mathbf{Q}_p$ by the same formula as for $\mathbf{Z}_p$. It appears that $\mathbf{Q}_p$ is a complete field (in particular locally compact Hausdorff) and that $\mathbf{Z}_p$ is an open subring.

The distance $d$ satisfies the “utrametric” inequality

$d(x,z) \leq sup\{d(x,y),d(y,z)\}$

## Properties

### As a field completion

Ostrowski's theorem implies there are precisely two kinds of completions of the rational numbers: the real numbers and the $p$-adic numbers.

###### Theorem

(Ostrowski)

Any non-trivial absolute value on the rational numbers is equivalent either to the standard real absolute value, or to the $p$-adic absolute value.

## References

The $p$-adic numbers had been introduced in

• Kurt Hensel, Über eine neue Begründung der Theorie der algebraischen Zahlen Jahresbericht der Deutschen Mathematiker-Vereinigung 6 (3): 83–88. (1897)

A standard reference is

A formalization in homotopy type theory and there in Coq is discussed in

$p$-adic differential equations are discussed in

p-adic homotopy theory is discussed in

Revised on February 26, 2014 23:09:25 by Anonymous Coward (78.52.172.141)