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p-adic number

pp-adic numbers

Idea

For pp any prime number, the pp-adic numbers p\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 pp-adic numbers form a non-archimedean field.

pp-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 pp-adic numbers also play a central role in the description of physics, see p-adic physics.

Definition

We first recall the definition and construction of the p-adic integers

and then consider

Recollection of the pp-adic integers

Let Z\mathbf{Z} be the ring of integers and for every q0q\neq 0, qZq\mathbf{Z} its ideal consisting of all integer multiples of qq, and Z/qZ\mathbf{Z}/q\mathbf{Z} the corresponding quotient, the ring of residues mod qq.

Let now pZ +p\in \mathbf{Z}_+ be a prime number. Then for any two positive integers nmn\geq m there is an inclusion p mZp nZp^m \mathbf{Z}\subset p^n\mathbf{Z} which induces the canonical homomorphism of quotients ϕ n,m:Z/p nZZ/p mZ\phi_{n,m}:\mathbf{Z}/p^n\mathbf{Z}\to \mathbf{Z}/p^m\mathbf{Z}. These homomorphism for all pairs nmn\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 Z p\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 Z p\mathbf{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\mathbf{Z}_p is the closed (hence compact) subspace of the cartesian product nZ/p nZ\prod_{n} \mathbf{Z}/p^n\mathbf{Z} of discrete topological spaces Z/p nZ\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)x = (...,x_n,...,x_2,x_1) with x np nZx_n\in p^n\mathbf{Z} and satisfying ϕ n,m(x n)=x m\phi_{n,m}(x_n) = x_m.

The kernel of the projection pr n:Z pZ/p nZpr_n: \mathbf{Z}_p\to\mathbf{Z}/p^n\mathbf{Z}, xx nx\mapsto x_n to the nn-th component (which is the corresponding projection of the limiting cone) is p nZ pZ pp^n\mathbf{Z}_p\subset\mathbf{Z}_p, i.e. the sequence

0Z pp nZ pZ/p nZ0 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 Z p/p nZ pZ/p nZ\mathbf{Z}_p/p^n\mathbf{Z}_p\cong \mathbf{Z}/p^n\mathbf{Z}.

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

Let UZ pU\subset\mathbf{Z}_p be the group of all invertible elements in Z p\mathbf{Z}_p. Then every element xZ px\in \mathbf{Z}_p can be uniquely written as s=up ns= up^n with n0n\geq 0 and uUu\in U. The correspondence xnx\mapsto n defines a discrete valuation v p:Z pZ{}v_p:\mathbf{Z}_p\to \mathbf{Z}\cup\{\infty\} called the pp-adic valuation and nn is said to be the pp-adic valuation of xx. Of course, v p(0)=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(xy), d(x,y) = e^{-v_p(x-y)},

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

The pp-adic numers proper

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

The distance dd satisfies the “utrametric” inequality

d(x,z)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 pp-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 pp-adic absolute value.

References

The pp-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

pp-adic differential equations are discussed in

p-adic homotopy theory is discussed in

Revised on April 7, 2014 08:22:07 by Urs Schreiber (88.128.80.110)