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For any prime number, the -adic numbers 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 -adic numbers form a non-archimedean field.
-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 -adic numbers also play a central role in the description of physics, see p-adic physics.
Let be the ring of integers and for every , its ideal consisting of all integer multiples of , and the corresponding quotient, the ring of residues mod .
Let now be a prime number. Then for any two positive integers there is an inclusion which induces the canonical homomorphism of quotients . These homomorphism for all pairs 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 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 has a natural topology as a profinite (Stone) space and it is in particular a compact Hausdorff topological ring. More concretely, is the closed (hence compact) subspace of the cartesian product of discrete topological spaces (which is by the Tihonov theorem compact Hausdorff) consisting of threads, i.e. sequences of the form with and satisfying .
The kernel of the projection , to the -th component (which is the corresponding projection of the limiting cone) is , i.e. the sequence
is an exact sequence of abelian groups, hence also .
An element in is invertible (and called a -adic unit) iff is not divisible by .
Let be the group of all invertible elements in . Then every element can be uniquely written as with and . The correspondence defines a discrete valuation called the -adic valuation and is said to be the -adic valuation of . Of course, as required by the axioms of valuation. The metric induced by the valuation is (up to equivalence) given by
ring is a complete metric space in that is a metric, and is dense in it.
The field of -adic numbers is the field of fractions of . The -adic valuation extends to a discrete valuation, also denoted on . Indeed, it is still true for all that they can be uniquely written in the form where (the same group as before), but now one needs to allow . One defines the metric on by the same formula as for . It appears that is a complete field (in particular locally compact Hausdorff) and that is an open subring.
The distance satisfies the “utrametric” inequality
As a field completion
Ostrowski's theorem implies there are precisely two kinds of completions of the rational numbers: the real numbers and the -adic numbers.
Any non-trivial absolute value on the rational numbers is equivalent either to the standard real absolute value, or to the -adic absolute value.
The -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
-adic differential equations are discussed in
p-adic homotopy theory is discussed in