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单词 PadicCanonicalForm
释义

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,

ξ=a-mp-m+a-m+1p-m+1++a0+a1p+a2p2+

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

-1=111111,
1=0001,
6.5=000110.1,
15=00110011001101.

(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

α=1000010001001011.10111.

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 division  1: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

|ξ|2=2-n.
Titlep-adic canonical form
Canonical namePadicCanonicalForm
Date of creation2013-03-22 14:13:37
Last modified on2013-03-22 14:13:37
Ownerpahio (2872)
Last modified bypahio (2872)
Numerical id66
Authorpahio (2872)
Entry typeExample
Classificationmsc 12F99
Related topicIntegralElement
Related topicUltrametricTriangleInequality
Related topicNonIsomorphicCompletionsOfMathbbQ
Related topicIdealsOfADiscreteValuationRingArePowersOfItsMaximalIdeal
Definesproper p-adic number
Definesdyadic number
Definesdyadic point
Defines2-adic fractional number
Defines2-adic integer
Defines2-adic valuation
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更新时间:2025/5/4 16:09:25