nLab
p-adic number

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 $Z$ be the ring of integers and for every $q\ne 0$, $qZ$ its ideal consisting of all integer multiples of $q$, and $Z/qZ$ the corresponding quotient, the ring of residues mod $q$.

Let now $p\in Z_{+}$ be a prime number. Then for any two positive integers $n\ge m$ there is an inclusion $p^{m}Z\subset p^{n}Z$ which induces the canonical homomorphism of quotients ${\varphi }_{n,m}:Z/p^{n}Z\to Z/p^{m}Z$. These homomorphism for all pairs $n\ge m$ 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 $Z_p$ 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 $Z_p$ has a natural topology as a profinite (Stone) space and it is in particular a compact Hausdorff topological ring. More concretely, $Z_p$ is the closed (hence compact) subspace of the cartesian product ${\prod }_{n}Z/p^{n}Z$ of discrete topological spaces $Z/p^{n}Z$ (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 $x_n\in p^{n}Z$ and satisfying ${\varphi }_{n,m}(x_n)=x_m$.

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

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

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

An element $u$ in $Z_p$ is invertible (and called a $p$-adic unit) iff $u$ is not divisible by $p$.

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

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

ring $Z_p$ is a complete metric space in that $d$ is a metric, and $Z$ is dense in it.

The field of $p$-adic numbers $Q_p$ is the field of fractions of $Z_p$. The $p$-adic valuation $v_p$ extends to a discrete valuation, also denoted $v_p$ on $Q_p$. Indeed, it is still true for all $x\in Q_p$ 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 $n\in Z$. One defines the metric on $Q_p$ by the same formula as for $Z_p$. It appears that $Q_p$ is a complete field (in particular locally compact Hausdorff) and that $Z_p$ is an open subring.

The distance $d$ satisfies the “utrametric” inequality

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

Applications

• p-adic physics

Related concepts

• carrying

• natural number, integer, rational number, algebraic number, Gaussian number, irrational number, real number

• p-adic integer

• p-adic analytic space, non-commutative analytic space

References

A standard reference is

• Jean-Pierre Serre, A course in arithmetic, Grad. Texts in Math. 7, Springer (1973)

$p$-adic differential equations are discussed in

• Kiran Kedlaya, $p$-adic differential equations (pdf, course notes)

• Gilles Cristol, Exposants $p$-adiques et solutions dans les couronnes (pdf)

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

• Álvaro Pelayo, Vladimir Voevodsky, Michael Warren, A preliminary univalent formalization of the p-adic numbers (arXiv:1302.1207)