p-adic canonical form

Every non-zero p-adic number (p is a positive rational prime number) can be uniquely written in canonical formMathworldPlanetmath, formally as a Laurent series,


where  m,  0akp-1  for all k’s, and at least one of the integers ak is positive.  In addition, we can write:  0=0+0p+0p2+

The field p of the p-adic numbers is the completion of the field with respect to its p-adic valuationMathworldPlanetmath (http://planetmath.org/PAdicValuation); thus may be thought the subfieldMathworldPlanetmath (prime subfieldMathworldPlanetmath) of p.  We can call the elements of  p  the proper p-adic numbers.

If, e.g.,  p=2,  we have the 2-adic or, according to G. W. Leibniz, dyadic numbers, for which every ak is 0 or 1.  In this case we can write the sum expression for ξ in the reverse and use the ordinary positional (http://planetmath.org/Base3) (i.e., dyadic) figure system (http://planetmath.org/Base3).  Then, for example, we have the rational numbers


(You may check the first by adding 1, and the last by multiplying by  5 = …000101.) All 2-adic rational numbers have periodic binary expansion (http://planetmath.org/DecimalExpansion).  Similarly as the decimal (http://planetmath.org/DecimalExpansion) (according to Leibniz: decadic) expansions of irrational real numbers are aperiodic, the proper 2-adic numbers also have aperiodic binary expansion, for example the 2-adic fractional number


The 2-adic fractional numbers have some bits “1” after the dyadic point “.” (in continental Europe: comma “,”), the 2-adic integers have none.  The 2-adic integers form a subring of the 2-adic field 2 such that 2 is the quotient field of this ring.

Every such 2-adic integer ε whose last bit is “1”, as  -3/7=11011011011, is a unit of this ring, because the division1:ε  clearly gives as quotient a integer (by the way, the divisions of the binary expansions in practice go from right to left and are very comfortable!).

Those integers ending in a “0” are non-units of the ring, and they apparently form the only maximal idealMathworldPlanetmath in the ring (which thus is a local ringMathworldPlanetmath).  This is a principal idealMathworldPlanetmath 𝔭, the generator of which may be taken  00010=10 (i.e., two).  Indeed, two is the only prime number of the ring, but it has infinitely many associatesMathworldPlanetmath, a kind of copies, namely all expansions of the form  10=ε10.  The only non-trivial ideals in the ring of 2-adic integers are  𝔭,𝔭2,𝔭3,  They have only 0 as common element.

All 2-adic non-zero integers are of the form ε2n where  n=0, 1, 2,.  The values  n=-1,-2,-3,  here would give non-integral, i.e. fractional 2-adic numbers.

If in the binary of an arbitrary 2-adic number, the last non-zero digit “1” corresponds to the power 2n, then the 2-adic valuation of the 2-adic number ξ is given by

Title p-adic canonical form
Canonical name PadicCanonicalForm
Date of creation 2013-03-22 14:13:37
Last modified on 2013-03-22 14:13:37
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 66
Author pahio (2872)
Entry type Example
Classification msc 12F99
Related topic IntegralElement
Related topic UltrametricTriangleInequality
Related topic NonIsomorphicCompletionsOfMathbbQ
Related topic IdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal
Defines proper p-adic number
Defines dyadic number
Defines dyadic point
Defines 2-adic fractional number
Defines 2-adic integer
Defines 2-adic valuation